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Copyright © 1996-2001 jsd
3 Airfoils and Airflow
- Have you heard how to make a small fortune in the
- Start with a large one.
3.1 Flow Patterns Near a Wing In this chapter I will explain a few things about how air behaves as
it flows past a wing. There will be lots of illustrations, such as
figure 3.1, produced by a wind-tunnel
simulation1 program that I wrote
for my computer. The wing is stationary in the middle of the wind
tunnel; air flows past it from left to right. A little ways upstream
of the wing (near the left edge of the figure) I have arranged a
number of smoke injectors. Seven of
them are on all the time, injecting thin streams of purple smoke. The
smoke is carried past the wing by the airflow, making visible
In addition, on a five-times closer vertical spacing, I inject
pulsed streamers. The smoke is turned on for 10
milliseconds out of every 20. In the figure, the blue smoke was
injected starting 70 milliseconds ago, the green smoke was injected
starting 50 milliseconds ago, the orange smoke was injected starting
30 milliseconds ago, and the red smoke was injected starting 10
milliseconds ago. The injection of the red smoke was ending just as
the snapshot was taken.
The set of all points that passed the injector array
at a given time defines a timeline. The right-hand edge
of the orange smoke is the ``30 millisecond'' timeline.
Figure 3.2 points out some important properties of
the airflow pattern. For one thing, we notice that the air just ahead
of the wing is moving not just left to right but also upward; this is
called upwash. Similarly, the air just aft of the wing is
moving not just left to right but also downward; this is called
downwash. Downwash behind the wing is relatively easy to
understand; the whole purpose of the wing is to impart some downward
motion to the air.
The upwash in front of the wing is a bit more interesting. As
discussed in section 3.6, air is a fluid, which means it can
exert pressure on itself as well as other things. The air pressure
strongly affects the air, even the air well in front of the wing.
Along the leading edge of the wing there is something called a
stagnation line, which is the dividing line between air that
flows over the top of the wing and air that flows under the bottom of
the wing. On an airplane, the stagnation line runs the length of the
wingspan, but since figure 3.2 shows only a cross
section of the wing, all we see of the stagnation line is a single
Another stagnation line runs spanwise along the trailing
edge. It marks the place where air that passed above the wing
rejoins air that passed below the wing.
We see that at moderate or high angles of attack, the forward
stagnation line is found well below and aft of the leading
edge of the wing. The air that meets the wing just above the
stagnation line will backtrack toward the nose of the airplane, flow
up over the leading edge, and then flow aft along the top of the wing.
Figure 3.3 introduces some additional useful concepts.
Since the air near the wing is flowing at all sorts of different
speeds and directions, the question arises of what is the ``true''
airspeed in the wind tunnel. The logical thing to do is to measure
the velocity of the free stream; that is, at a point well
upstream, before it has been disturbed by the wing.
The pulsed streamers give us a lot of information. Regions where the
pulsed streamers have been stretched out are high velocity regions.
This is pretty easy to see; each pulsed streamer lasts exactly 10
milliseconds, so if it covers a long distance in that time it must be
moving quickly. The maximum velocity produced by this wing at this
angle of attack is about twice the free-stream velocity.
Airfoils can be very effective at speeding up the air.
Conversely, regions where the pulsed streamers cover
a small distance in those 10 milliseconds must be low-velocity
regions. The minimum velocity is zero. That occurs near the
front and rear stagnation lines.
The relative wind vanishes on the stagnation lines.
A small bug walking on the wing of an airplane in flight could
walk along the stagnation line without feeling any wind.2
Stream lines have a remarkable property: the air
can never cross a stream line. That is because of the way the stream
lines were defined: by the smoke. If any air tried to flow past a
point where the smoke was, it would carry the smoke with it.
Therefore a particular parcel of air bounded by a pair of stream lines
(above and below) and a pair of timelines (front and rear) never loses
its identity. It can change shape, but it cannot mix with another
Another thing we should
notice is that in low velocity regions, the stream lines are
farther apart from each other. This is no accident. At
reasonable airspeeds, the wing doesn't push or pull on the air hard
enough to change its density significantly (see
section 3.4.3 for more on this). Therefore the air
parcels mentioned in the previous paragraph do not change in area when
they change their shape. In one region, we have a long, skinny parcel
of air flowing past a particular point at a high velocity. (If the
same amount of fluid flows through a smaller region, it must be
flowing faster.) In another region, we have a short fat parcel
flowing by at a low velocity.
The most remarkable thing about this figure is that the blue smoke
that passed slightly above the wing got to the trailing edge 10 or 15
milliseconds earlier than the corresponding smoke that passed
slightly below the wing.
This is not a mistake. Indeed, we shall see in section 3.10.3
that if this were not true, it would be impossible for the wing to
This may come as a shock to many readers, because all sorts of
standard references claim that the air is somehow required to pass
above and below the wing in the same amount of time. I have seen this
erroneous statement in elementary-school textbooks, advanced physics
textbooks, encyclopedias, and well-regarded pilot training handbooks.
Bear with me for a moment, and I'll convince you that figure 3.3 tells the true story.
First, I must convince you that there is no law of
physics that prevents one bit of fluid from being delayed relative
Consider the scenario depicted in figure 3.4. A river of water
is flowing left to right. Using a piece of garden hose, I siphon some
water out of the river, let it waste some time going through several
feet of coiled-up hose, and then return it to the river. The water
that went through the hose will be delayed. The delayed parcel of
water will never catch up with its former neighbors; it will not even
try to catch up.
Note that delaying the water did not require
compressing the water, nor did it require
The same story applies to air. Air flowing past an obstacle will be
delayed. In fact, air that comes arbitrarily close to a stagnation
line will be delayed an
arbitrarily long time. The air molecules just hang around in the
vicinity of the stagnation line, like the proverbial mule midway
between two bales of hay, unable to decide which alternative to
choose. This delay occurs even when the wing is producing zero lift,
as shown in the top panel of figure 3.5.
You can see that in all cases the air that hit the stagnation line
dead-on — the middle blue streamer — never makes it to the
trailing edge in any of these figures.
When a wing is not producing lift4 it is just
a slight obstacle to the airflow. Air passing near the wing is
slightly delayed, but that's about all. Air that passed slightly
above the wing is delayed about the same amount as the corresponding
air that passed below the wing.
When the wing is producing lift, the airflow patterns become much more
interesting, as you can see from the other panels of figure 3.5.
Air that passes above the wing reaches the trailing edge substantially
earlier than it would have if the wing had not been producing lift
(except for a tiny parcel of air, invisible in the picture, that just
barely missed the stagnation line). Air that passes below the wing is
substantially delayed. These effects extend for quite a distance
above and below the wing.
A wing (when it is producing lift) is amazingly effective
at speeding up the air above
it. Even though the air that passes above the wing has a longer
path, it gets to the back earlier than the corresponding
air that passes below the wing.
The change in speed is only temporary. As the air reaches the
trailing edge and thereafter, it quickly returns to its original,
free-stream velocity (plus a slight downward component). This can
been seen in the figures, such as figure 3.3 — the
spacing between successive smoke pulses returns to its original value.
The change in relative position is permanent. If we follow the air
far downstream of the wing, we find that the air that passed below the
wing will never catch up with the corresponding air that passed above
the wing. It will not even try to catch up.
3.2 Pressure Patterns Near a WingFigure 3.6 is a contour plot that shows what the pressure
is doing in the vicinity of the wing. All pressures will be measured
relative to the ambient atmospheric pressure in the free stream. The
blue-shaded regions indicate suction, i.e. negative pressure relative
to ambient, while the red-shaded regions indicate positive pressure
relative to ambient. The dividing line between pressure and suction
is also indicated in the figure.
The pressure and suction created by the wing are conveniently measured
in multiples of the dynamic pressure.5 It is usually represented by the symbol Q.
For a typical general-aviation flight situation, Q is about half a
pound per square inch. The maximum positive pressure on the airfoil
is exactly equal to Q; this occurs right at the stagnation
lines.6 The maximum suction
depends on the angle of attack, and on the detailed shape of the
airfoil; for the situation in figure 3.6 the max suction is
just over 0.8 Q. Each contour in the figure represents exactly 0.2
Q (roughly 0.1 psi).
There is a lot we can learn from studying this figure. For one thing,
we see that the front quarter or so of the wing does half of the
lifting. Another thing to notice is that suction acting on the top of
the wing is vastly more important than pressure acting on the bottom
of the wing. In figure 3.6, the wing is flying at an angle
of attack of 3 degrees, a reasonable ``cruise'' value.
At this angle of attack, there is almost no high pressure on the
bottom of the wing; indeed there is mostly suction there. The
only reason the wing can support the weight of the airplane is that
there is more suction on the top of the wing. (There is a tiny
amount of positive pressure on the rear portion of the bottom surface,
but the fact remains that suction above the wing does more than 100%
of the job of lifting the airplane.)7
Once again, this pressure pattern would be really hard to explain in
terms of bullets bouncing off the wing. Remember, the air is a fluid.
It has a well-defined pressure everywhere in space. When this
pressure field meets the wing, it exerts a force: pressure times area
At higher angles of attack, above-atmospheric pressure does develop
below the wing, but it is always less pronounced than the
below-atmospheric pressure above the wing.
3.3 Stream Line CurvatureFigure 3.7 shows what happens near the wing when we change the
angle of attack. You can see that as the velocity changes, the
pressure changes also.
It turns out that given the velocity field, it is rather
straightforward to calculate the pressure field. Indeed there are two
ways to do this; we discuss one of them here, and the other in
We know that air has mass. Moving air has momentum. If the air
parcel follows a curved path, there must be a net force on it, as
required by Newton's laws.8
Pressure alone does not make a net force; you need a pressure
difference so that one side of the air parcel is being pressed
harder than the other. Therefore the rule is this: If at any place
the stream lines are curved, the pressure at nearby places is
You can see in the figures that tightly-curved streamlines correspond
to big pressure gradients and vice versa.
If you want to know the pressure everywhere, you can start somewhere
and just add up all the changes as you move from place to place to
place. This is mathematically tedious, but it works. It works even
in situations where Bernoulli's principle isn't immediately applicable.
3.4 Bernoulli's PrincipleWe now discuss a second way in which pressure is related to velocity,
namely Bernoulli's principle. In situations where this principle can
be applied (which includes most situations), this is by far the
slickest way to do it.
Bernoulli's principle is derived from the law of conservation of
energy. It involves the kinetic energy of moving air and the potential energy stored in the ``springiness'' of the air.
Just as energy can be stored in a wound-up spring, energy is stored in
Pressure, denoted P, is (by definition) a force per unit area,
which is the same thing as an energy per unit volume:9
Meanwhile, moving air contains kinetic energy
just like any other moving object:
= Potential Energy per volume
where v is the local velocity, and r (the Greek letter
``rho'') is the density, i.e. the mass per unit volume.
= Kinetic Energy per volume
Note: In your browser, the following should look like Greek letters:
``gp''. If they look like ``gamma'' and ``pi'' then all is
well. If they look like Roman letters such as ``g'' and ``p'' then your
browser has not properly loaded the symbol fonts. To fix this please
refer to the font-fixing notes.
Combining these, we conclude:
Next, we make the approximation that we can ignore non-mechanical
forms of energy (such as chemical reactions or heat produced by
friction), and that we are not adding energy to the air using pumps,
pistons, or whatever. Then, using the law that total energy cannot
change (see chapter 1), we conclude that a given air
parcel's mechanical energy remains constant as it flows past the wing.
= Mechanical Energy per volume
Now, if the right-hand side of equation 3.3 is a constant, it
tells us that whenever a given parcel of air increases its velocity,
it must decrease its pressure, and vice versa. This relationship is
called Bernoulli's principle.
Higher velocity means lower pressure, and vice versa
(assuming constant mechanical energy).
Oftentimes10 it turns out that all the air
parcels start out with the same mechanical energy. In such a case we
can even make a Bernoulli-like statement comparing different
parcels of air: Any fast-moving air must have lower pressure than any
slow-moving air with the same mechanical energy.
Bernoulli's principle cannot be trusted if processes
other than kinetic energy and pressure energy are important.
In particular, in the ``boundary layer'' very near the
surface of a wing, energy is constantly being dissipated (converted
to heat) by friction. Fortunately, the boundary layer is usually
very thin (except near the stall), and if we ignore it entirely
Bernoulli's principle gives essentially the right answer.
It makes sense to measure the local velocity (lower-case
v) at each point as a multiple of the free-stream velocity
(capital V) since they vary in proportion to each other.
Similarly it makes sense to measure relative pressures in terms
of the dynamic pressure:
which is always small compared to atmospheric pressure (assuming V
is small compared to the speed of sound). The pressure versus
velocity relationship is shown graphically in figure 3.8.
The highest possible pressure (corresponding to completely stopped
air) is one Q above atmospheric, while fast-moving air can have
pressure several Q below atmospheric.
It doesn't matter whether we measure P as
an absolute pressure or as a relative pressure (relative to atmospheric).
If you change from absolute to relative pressure it just shifts
both sides of Bernoulli's equation by a constant, and the new
value (just as before) remains constant as the air parcel flows
past the wing. Similarly, if we use relative pressure in figure 3.8, we can drop the word ``Atm'' from the
pressure axis and just speak of ``positive one Q''
and ``negative two Q'' — keeping in mind that all
the pressures are only slightly above or below one atmosphere.
Bernoulli's principle allows us to understand why there is a positive
pressure bubble right at the trailing edge of the wing (which is the
last place you would expect if you thought of the air as a bunch of
bullets). The air at the stagnation line is the slowest-moving air in
the whole system; it is not moving at all. It has the highest
possible pressure, namely Atm + Q.
As we saw in the bottom panel of figure 3.7, at high angles of
attack a wing is extremely effective at speeding up the air above the
wing and retarding the air below the wing. The maximum local velocity
above the wing can be more than twice the free-stream velocity.
This creates a negative pressure (suction) of more than 3 Q.
3.4.2 Altimeters; Static versus Stagnation PressureConsider the following line of reasoning:
The answer has to do with the notion of ``lower'' pressure. You have
to ask, lower than what? Indeed the pressure there is
lower than the mechanical energy (per unit volume) of the air.
However, in your reference
frame, the mechanical energy of the air is
1 Atm + 1 Q.
When we subtract
from that, we see that the pressure in the static port is just equal
to atmospheric. Therefore the altimeter gets the right answer,
independent of airspeed.
airplane's altimeter operates by measuring the pressure at the
- The static port is oriented sideways to the airflow, at a point
chosen so that the air flows past with a local velocity just equal to
the free-stream velocity.
- In accordance with Bernoulli's principle, this velocity must be
associated with a ``lower'' pressure there.
- You might think this lower pressure would cause huge errors in
the altimeter, depending on airspeed. In fact, though, there are no
such errors. The question is, why not?
Another way of saying it is that the air near the static port has
of potential energy (pressure) and
of kinetic energy.
In contrast, the air in the Pitot tube has the same mechanical energy,
1 Atm + 1 Q,
but it is all in the form of potential energy since (in your reference
frame) it has no kinetic energy.
The mechanical energy per unit volume is officially called the
stagnation pressure, since it is the pressure that you observe
in the Pitot tube or any other place where the air is stagnant,
i.e. where the local velocity v is zero (relative to the airplane).
In ordinary language ``static'' and ``stagnant'' mean almost the same
thing, but in aerodynamics they designate two very different concepts.
The static pressure is the pressure you would measure in the
reference frame of the air, for instance if you were in a balloon
comoving with the free stream. As you increase your airspeed, the
stagnation pressure goes up, but the static pressure does not.
Also: we can contrast this with what happens in a carburetor.
There is no change of reference frames, so the mechanical energy (per
unit volume) remains
The high-speed air in the throat of the Venturi has a pressure
below the ambient atmospheric pressure.
3.4.3 CompressibilityFirst, a bit of terminology:
Non-experts may not make much distinction between a ``pressurized''
fluid and a ``compressed'' fluid, but in the engineering literature
there is a world of difference between the two concepts.
Pressure denotes a force per unit area.
- Compressibility denotes a change in density in response
Every substance on earth is compressible — be it air, water, cast
iron, or anything else. It must increase its density when you apply
pressure; otherwise there would be no way to balance the energy
However, changes in density are not very important to understanding
how wings work, as long as the airspeed is not near or above the speed
of sound. Typical general aviation airspeeds correspond to Mach 0.2
or 0.3 or thereabouts (even when we account for the fact that the wing
speeds up the air locally), and at those speeds the density never
changes more than a few percent.
For an ideal gas, density is proportional to pressure, so you may be
wondering why pressure-changes are important but density-changes are
not. Here's why:
To say it again: Flight depends directly on total density but
not directly on total atmospheric pressure, just pressure
Lift depends on a pressure difference between the top and
bottom of the wing. Similarly pressure drag depends on pressure
differences. Therefore the relevant differential pressures are
zero plus important terms proportional to
- Meanwhile, the density is some big number plus or minus
unimportant terms proportional to
Many books say the air is ``incompressible'' in the subsonic
regime. That's bizarrely misleading. In fact, when those books use
the words ``incompressible flow'' it generally means that the density
undergoes only small-percentage changes. This has got nothing to do
with whether the fluid has a high or low compressibility. The real
explanation is that the density-changes are small because the
pressure-changes are small compared to the total atmospheric pressure.
Similarly, many books say that equation 3.3 only applies to an
``incompressible'' fluid. Again, that's
bizarrely misleading. Here's the real story:
Here is Bernoulli's equation including the second-order term. I have
rewritten it in terms of energy per mass (rather than energy per
volume), to make it clear that compression doesn't matter, since a
parcel's mass doesn't change even if its volume changes:
Compressibility specifies to first order how density depends on
pressure. Equation 3.1 specifies to first order how the energy
depends on pressure. It already accounts for the effects of
compressibility and all other first-order quantities. Therefore
equation 3.3 is valid whenever the pressure-changes are a small
percentage of the total pressure, regardless of compressibility.
- At high airspeeds, the pressure changes are bigger, and you need
a more sophisticated form of Bernoulli's equation. As shown below, it
is straightforward to include second-order terms — which, by the
way, don't depend on compressibility, either. Indeed you can use the
full equation of state, to derive Bernoulli's equation in a form
that is valid even for large-percentage changes in pressure. See
reference 2, page 29, equation 11.
where r0 is the
density of air at atmospheric pressure, and
where g (gamma) is a constant that appears in the
equation of state
for the fluid. Its value ranges from 1.666 for helium, to 1.4 for
air, to 1.0 for cool liquid water. It's ironic that the correction is
actually smaller for air (which has a high compressibility) than it is
for water (which has a much lower compressibility). So don't let
anybody tell you that Bernoulli's principle can't account for
compressibility. It already does, even in its simplest form.
= Mechanical Energy per mass
3.5 Stall Warning DevicesWe are now in a position to understand how stall warning devices
work. There are two types of stall-warning devices commonly used on
light aircraft. The first type (used on most Pipers, Mooneys, and
Beechcraft) uses a small vane mounted slightly below and aft of the
leading edge of the wing as shown in the left panel of
figure 3.9. The warning is actuated when the vane is blown
and forward. At low angles of attack (e.g. cruise) the stagnation
line is forward of the vane, so the vane gets blown backward and
everybody is happy. As the angle of attack increases, the stagnation
line moves farther and farther aft underneath
the wing. When it has moved farther aft than the vane, the air will
blow the vane forward and upward and the stall warning will be
The second type of stall-warning device (used on the Cessna 152, 172,
and some others, not including the 182) operates on a different
principle. It is sensitive to suction at the surface rather than flow
along the surface. It is positioned just below the leading edge of
the wing, as indicated in the right panel of figure 3.9.
At low angles of attack, the leading edge is a low-velocity,
high-pressure region; at high angles of attack it becomes a
high-velocity, low-pressure region. When the low-pressure region
extends far enough down around the leading edge, it will suck air out
of the opening. The air flows through a harmonica reed, producing an
Note that neither device actually detects the stall.
Each one really just measures angle of attack. It is designed
to give you a warning a few degrees before the wing reaches
the angle of attack where the stall is expected. Of course if
there is something wrong, such as frost on the wing (see
section 3.13), the wing will stall at a lower-than-expected angle
of attack, and you will get no warning from the so-called stall
3.6 Air Is A Fluid, Not A Bunch of BulletsWe all know that at the submicroscopic level, air consists of
particles, namely molecules
of nitrogen, oxygen, water, and various other substances. Starting
from the properties of these molecules and their interactions, it is
possible to calculate macroscopic properties such as pressure,
velocity, viscosity, speed of sound, et cetera.
However, for ordinary purposes such as understanding how wings work,
you can pretty much forget about the individual particles, since the
relevant information is well summarized by the macroscopic properties
of the fluid. This is
called the hydrodynamic approximation.
In fact, when people try to think about the individual
particles, it is a common mistake to overestimate the size of
the particles and to underestimate the importance of the interactions
If you erroneously imagine that air particles are large and
non-interacting, perhaps like the bullets shown
in figure 3.10, you will never understand how wings work.
Consider the following comparisons. There is only one important thing
bullets and air molecules have in common:
Bullets hit the bottom of the wing, transferring upward momentum
Similarly, air molecules hit the bottom of the wing, transferring
upward momentum to it.
Otherwise, all the important parts of the story are different:
No bullets hit the top of the wing.
Air pressure on top of the wing is only a few percent lower than the
pressure on the bottom.
The shape of the top of the wing doesn't matter to the bullets.
The shape of the top of the wing is crucial. A spoiler at
location ``X'' in figure 3.10 could easily double the drag of
the entire airplane.
The bullets don't hit each other, and
even if they did, it wouldn't affect lift production.
Each air molecule collides with one or another of its neighbors
10,000,000,000 times per second. This is crucial.
Each bullet weighs a few grams.
Each nitrogen molecule weighs
Bullets that pass above or below the wing are undeflected.
The wing creates a pressure field that strongly deflects
even far-away bits of fluid.
Bullets could not possibly knock a stall-warning vane forward.
Fluid flow nicely explains how such a vane gets blown
forward and upward. See section 3.5.
The list goes on and on, but you get the idea. Interactions between
air molecules are a big part of the story. It is a much better
approximation to think of the air as a continuous fluid
than as a bunch of bullet-like particles.
3.7 Other FallaciesYou may have heard stories that try to use the Coanda effect or
the teaspoon effect to explain how wings produce lift. These
stories are completely fallacious, as discussed in section 18.4.4 and section 18.4.3.
There are dozens of other fallacies besides. It is beyond the scope
of this book to discuss them, or even to catalog them all.
3.8 Inverted Flight, Cambered vs. Symmetric AirfoilsAlmost everybody has been told that an airfoil produces
lift because it is curved on top and flat on the
bottom. But aren't you also aware that airshow pilots routinely
fly for extended periods of time upside
down? Doesn't that make you suspicious that there might be something
wrong with the story about curved on top and flat on the bottom?
Here is a list of things you need in an airplane
intended for upside-down flight:
You will notice that changing the cross-sectional
shape of the wing is not on this list. As shown in figure 3.11,
an ordinary wing flies just fine inverted. It looks rather peculiar
flying with the flat surface on top and the rounded surface on
the bottom, but it works.
You need super-duper
seatbelts to keep the pilot from flopping around.
- You need to make sure the airframe is strong
enough to withstand extra stress, including stress in new directions.
- You need to make sure that the fuel, engine oil,
and battery acid stay where they are supposed to be.
The common misconception that wings must be curved
on top and flat on the bottom is related to the previously-discussed
misconception that the air is required to pass above and below
the wing in equal amounts of time. In fact, an upside-down wing
produces lift by exactly the same principle as a rightside-up
To help us discuss airfoil shapes, figure 3.12
illustrates some useful terminology.
A symmetric airfoil, where the top surface is a mirror image of
the bottom surface, has zero camber. The airflow and pressure
patterns for such an airfoil are shown in figure 3.13.
The chord line is the straight line drawn from the
leading edge to the trailing edge.
- The term camber in
general means ``bend''. If you want to quantify the amount of
camber, draw a curved line from the leading edge to the trailing edge,
staying always halfway between the upper surface and the lower
surface; this is called the mean camber line. The maximum
difference between this and the chord line is the amount of camber.
It can be expressed as a distance or (more commonly) as a percentage
of the chord length.
This figure could be considered the side view of
a symmetric wing, or the top view of a rudder. Rudders are airfoils,
too, and work by the same principles.
At small angles of attack, a symmetric airfoil works
better than a highly cambered airfoil. Conversely, at high angles
of attack, a cambered airfoil works better than the corresponding
symmetric airfoil. An example of this is shown in figure 3.14.
The airfoil designated ``631-012'' is symmetric, while
the airfoil designated ``631-412'' airfoil is cambered;
otherwise the two are identical.11 At any normal angle of attack (up to
about 12 degrees), the two airfoils produce virtually identical
amounts of lift. Beyond that point the cambered airfoil has a
big advantage because it does not stall until a much higher relative
angle of attack. As a consequence, its maximum coefficient of
lift is much greater.
At high angles of attack, the leading edge of a cambered
wing will slice into the wind at less of an angle compared to
the corresponding symmetric wing. This doesn't prove anything,
but it provides an intuitive feeling for why the cambered wing
has more resistance to stalling.
The amount of camber on a typical modern airfoil
is only 1 or 2 percent — obviously not crucial. One reason wings
are not more cambered is that any increase would require the bottom
surface to be concave — which would be a pain to manufacture.
Another reason is that large camber is only really beneficial
near the stall — i.e. for takeoff and landing, and it suffices
to create lots of camber by extending the flaps when needed.
Reverse camber is clearly a bad idea (since it causes
earlier stall) so aircraft that are expected to perform well upside
down (e.g. Pitts or Decathlon) have symmetric (zero-camber) airfoils.
We have seen that under ordinary conditions, the
amount of lift produced by a wing depends on the angle of attack,
but hardly depends at all on the amount of camber. This makes
sense. In fact, the airplane would be unflyable if the coefficient
of lift were determined solely by the shape of the wing. Since
the amount of camber doesn't often change in flight, there would
be no way to change the coefficient of lift. The airplane could
only support its weight at one special airspeed, and would be
unstable and uncontrollable. In reality, the pilot (and the trim
system) continually regulate the amount of lift by regulating
the all-important angle of attack; see chapter 2 and chapter 6.
3.9 Thin WingsThe wing used on the Wright brothers' first airplane is thin,
highly cambered, and quite concave on the bottom. This is shown in
figure 3.15. There is no significant difference between the
top surface and the bottom surface — same length, same curvature.
Still, the wing produces lift, using the same lift-producing principle
as any other airfoil. This should further dispel the notion that
wings produce lift because of a difference in length between the upper
and lower surfaces.
Similar remarks apply to the sail of a sailboat. It is a very thin
wing, oriented more-or-less vertically, producing sideways lift.
Even a thin flat object such as a barn door will produce lift,
if the wind strikes it at an appropriate angle of attack. The airflow
pattern (somewhat idealized) for a barn door (or the wing on a
dime-store balsa glider) is shown in figure 3.16. Once again,
the lift-producing mechanism is the same.
3.10.1 Visualizing the circulationYou may be wondering whether the flow patterns shown
in figure 3.16 or the earlier figures are the only ones
allowed by the laws of hydrodynamics. The answer is: almost,
but not quite. Figure 3.17 shows the barn door operating
with the same angle of attack (and the same airspeed) as in figure 3.16, but the airflow pattern is different.
The new airflow pattern (figure 3.17) is
highly symmetric. I have deleted the timing information, to make
it clear that the stream lines are unchanged if you flip the figure
right/left and top/bottom. The front stagnation line is a certain
distance behind the leading edge; the rear stagnation line is
the same distance ahead of the trailing edge. This airflow pattern
produces no lift. (There will be a lot of torque — the so-called Rayleigh torque — but no
The difference between these figures is circulation
— figure 3.16 has circulation while figure 3.17
does not. (Figure 3.19 is the same as figure 3.16
without the timing information.)
To understand circulation and its effects, first
imagine an airplane with barn-door wings, parked on the ramp on
a day with no wind. Then imagine stirring the air with a paddle,
setting up a circulatory flow pattern, flowing nose-to-tail over
the top of the wing and tail-to-nose under the bottom (clockwise
in this figure). This is the flow pattern for pure circulation,
as shown in figure 3.18. Then imagine that a headwind
springs up (left to right in the figure). At each point in space,
the velocity fields will add. The circulatory flow and the wind
will add above the wing, producing high velocity and low
pressure there. The circulatory flow will partially cancel
the wind below the wing, producing low velocity and high pressure
If we take the noncirculatory left-to-right flow in figure 3.17 and add various amounts of circulation, we can generate
all the flow patterns consistent with the laws of hydrodynamics
— including the actual natural airflow shown in figure 3.16
and figure 3.19.12
There is nothing special about barn doors; real airfoils
have analogous airflow patterns, as shown in figure 3.20,
figure 3.21, and figure 3.22.
: Unnatural Airflow — Angle of Attack
but No Circulation
If you suddenly accelerate a wing from a standing
start, the initial airflow pattern will be noncirculatory, as
shown in figure 3.20. Fortunately for us, the air
absolutely hates this airflow pattern, and by the time the wing
has traveled a short distance (a couple of chord-lengths or so)
it develops enough circulation to produce the normal airflow
pattern shown in figure 3.22.
3.10.2 How Much Circulation? The Kutta ConditionIn real flight situations, precisely enough circulation
will be established so that the rear stagnation line is right
at the trailing edge, so no air needs to turn the corner there.
Of course, the circulation that cancels the flow around the trailing
edge more or less doubles the flow around the leading edge.
The general rule — called the Kutta condition —
is that the air hates to turn the corner at a sharp trailing edge. To
a first approxmation, the air hates to turn the corner at any
sharp edge, because the high velocity there creates a lot of
friction. For ordinary wings, that's all we need to know, because the
trailing edge is the only sharp edge.
The funny thing is that the trailing edge is sharp, an airfoil will
work even if the leading edge is sharp, too. This explains why
dime-store balsa-wood gliders work, even with sharp leading edges.
It is a bit of a mystery
why the air hates turning a corner at the trailing edge, and doesn't
mind so much turning a sharp corner at the leading edge — but
that's the way it is.13 This is related to the well-known fact
that blowing is different from sucking. (Even though you can blow out
a candle from more than a foot away, you cannot suck out a candle from
more than an inch or two away.) In any case, the rule is:
The air wants to flow cleanly off the trailing edge.
As the angle of attack increases, the amount of circulation needed
to meet the Kutta condition increases.
Here is a nice, direct way of demonstrating the Kutta condition:
At a safe altitude, start with the airplane14 in the clean
configuration in level flight, a couple of knots above the speed where
the stall warning horn comes on.
Maintaining constant pitch attitude
and maintaining level flight, extend the flaps. The stall warning
horn will come on. There is no need to stall the airplane; the
warning horn itself makes the point.
This demonstration makes it clear that the flap (which is at the
back of the wing) is having a big effect on the airflow around
the entire wing, including the stall-warning detector (which is
near the front).
Extending the flaps (while maintaining constant pitch and constant
direction of flight) increases the angle of attack. This increases
the circulation, which trips the stall-warning detector as described
in section 3.5.
3.10.3 How Much Lift? The Kutta-Zhukovsky TheoremHere is a beautifully simple and powerful result: The lift is equal to
the airspeed, times the circulation, times the density of the air,
times the span of the wing. This is called the Kutta-Zhukovsky
Lift = airspeed × circulation
× density × span
Since circulation is proportional to the coefficient of lift and
to the airspeed, this new notion is consistent with our previous
knowledge that the lift should be proportional to the coefficient
of lift times airspeed squared.
You can look at a velocity field and visualize the circulation.
In figure 3.23, the vertical black line shows where
the 70 millisecond timeline would have been if the wing had been
completely absent. The actual 70 millisecond timeline is given
by the right-hand edge of the blue streamers.
: Circulation Advances Upper & Retards
Because of the circulatory contribution to the velocity, the streamers
above the wing are at a relatively advanced position, while the
streamers below the wing are at a relatively retarded position.
If you refer back to figure 3.7, you can see that circulation
is proportional to angle of attack. In particular, note that
when the airfoil is not producing lift there is no circulation
— the upper streamers are not advanced relative to the lower streamers.
The same thing can be seen by comparing figure 3.20
to figure 3.22 — when there is no circulation the upper
streamers are not advanced relative to the lower streamers.
3.10.4 Quantifying the CirculationCirculation can be measured, according to the following procedure.
Set up an imaginary loop around the wing. Go around the loop
clockwise, dividing it into a large number of small segments. For
each segment, multiply the length of that segment times the speed of
the air along the direction of the loop at that point. (If the
airflow direction is opposite to the direction of the loop, the
product will be negative.) Add up all the products. The total
velocity-times-length will be the circulation. This is the
Interestingly, the answer is essentially independent
of the size and shape of the loop.15 For instance, if you go farther away, the velocity will
be lower but the loop will be longer, so the velocity-times-length
will be unchanged.
3.11 Mechanically-Induced CirculationThere is a widely-held misconception that it is the
velocity relative to the skin of the wing that produces
lift. This causes no end of confusion.
Remember that the air has a well defined velocity
and pressure everywhere, not just at the surface of the wing.
You can go anywhere near or far from the wing with a windmill
and a pressure gauge and measure the velocity and pressure. The
circulatory flow set up by the wing creates low pressure in a
huge region extending far above the wing. The velocity at each
point determines the pressure at that point.
The circulation near a wing is normally set up by
the interaction of the wind with the shape of the wing. But there
are other ways of setting up circulatory flow. In figure 3.24,
the wings are not airfoil-shaped but paddle-shaped. By rotating
the paddle-wings we
can set up a circulatory airflow pattern by brute force.
Bernoulli's principle would apply point by point
in the air near the wing, creating low pressure that would pull
up on the wings, even though the air near the wing would have
no velocity relative to the wing since it would be ``stuck''
between the vanes of the paddle. The Kutta-Zhukovsky theorem
would apply, as stated above: lift is equal to the airspeed, times
the circulation, times the density of the air, times the span
of the wing.
This phenomenon — creating the circulation needed for lift by
mechanically stirring the air — is called the Magnus effect.
The airplane in figure 3.24 would have definite
controllability problems, since the notion of angle of attack
would not exist (see chapter 2 and chapter 6).
The concept, though, is not as ridiculous as might seem. The
famous aerodynamicist Flettner once built a ship that ``sailed'' all the way across the Atlantic
using huge rotating cylinders rather than sails to catch the wind.
Also, it is easier than you might think to demonstrate
this important concept. You don't need four vanes on the rotating
paddle; a single flat surface will do. A business card works
fairly well. Drop the card from shoulder height, with its long
axis horizontal. As you release it, give it a little bit of backspin
around the long axis. It will fly surprisingly well; the lift-to-drag
ratio is not enormous, but it is not zero either. The motion
is depicted in figure 3.25.
: Fluttering Card — Lift Created by Circulation
You can improve the performance by giving the wing
a finer aspect ratio (more span and/or less chord). I once took
a manila folder and cut out several pieces an inch wide and 11
inches long; they work great.
As an experiment, try giving the wing the wrong direction
of circulation (i.e. topspin) as you release it. What do you
think will happen?
I strongly urge you to try this demonstration yourself.
It will improve your intuition about the relationship of circulation
We can use these ideas to understand some (but not all) of the
aerodynamics of tennis balls and similar objects.
As portrayed in figure 3.26, if a ball is hit with a lot of
backspin, the surface of the spinning ball will create the
necessary to produce lift, and it will be a ``floater''. Conversely,
the classic ``smash'' involves topspin, which produces negative lift,
causing the ball to ``fly'' into the ground faster than it would under
the influence of gravity alone. Similar words apply to leftward and
rightward curve balls.
To get even close to the right answer, we must ask
where the relative wind is fast or slow, relative to the center
of the ball — not relative to the rotating surface of the ball.
Remember that the fluid has a velocity and a pressure everywhere,
not just at the surface of the ball. Air moving past a surface
is what creates drag, not lift. Bernoulli says that when
an air parcel accelerates or decelerates, it exchanges its kinetic
energy (airspeed) for potential energy (pressure). For the floater,
the circulatory flow created by the backspin combines with the
free-stream flow created by the ball's forward motion to create
high-velocity, low-pressure air above the ball — that is, lift.
This simple picture of mechanically-induced circulation
applies best to balls that have evenly-distributed roughness.
Cricket balls are in a different category, since they have a
prominent equatorial seam. If you spin-stabilize the orientation
of the seam, and fly the seam at an ``angle of attack'',
airflow over the seam causes extra turbulence which promotes attached
flow on one side of the ball. See section 18.3
for some discussion of attached versus separated flow. Such effects
can overwhelm the mechanically-induced circulation.
To really understand flying balls or cylinders, you
would need to account for the direct effect of spin on circulation,
the effect of spin on separation, the effect of seams on separation,
et cetera. That would go beyond the scope of this book. A wing
is actually easier to understand.
3.12 Lift Requires Circulation & Vortices
A vortex is a bunch of air circulating around itself. The axis
around which the air is rotating is called a vortex line. It is
mathematically impossible for a vortex line
to have loose ends. A smoke ring is an example of a vortex. It
closes on itself so it has no loose ends.
The circulation necessary to produce lift can attributed to a
bound vortex line. It binds to
the wing and travels with the airplane. The question arises, what
happens to this vortex line at the wingtips?
The answer is that the vortex spills off each wingtip. Each wing
forms a trailing vortex (also called
wake vortex) that extends for miles behind the airplane. These
trailing vortices constitute the continuation of the bound vortex.
See figure 3.27. Far behind the airplane, possibly all the
way back at the place where the plane left ground effect, the two
trailing vortices join up to form an unbroken vortex line.
The air rotates around the vortex line in the direction
indicated in the figure. We know that the airplane, in order
to support its weight, has to yank down on the air. The air that
has been visited by the airplane will have a descending motion
relative to the rest of the air. The trailing vortices mark the
boundary of this region of descending air.
It doesn't matter
whether you consider the vorticity to be the cause or the effect
of the descending air — you can't have one without the other.
Lift must equal weight times load factor, and we can't easily change
the weight, or the air density, or the wingspan. Therefore, when the
airplane flies at a low airspeed, it must generate lots of
* Winglets, etc.It is a common misconception that the wingtip vortices
are somehow associated with unnecessary spanwise flow, and that
they can be eliminated using fences, winglets,
et cetera. The reality is that the vortices are completely necessary;
you cannot produce lift without producing vortices. By fiddling
with the shape of the wing the designer can control where
along the span the vortices are shed, but there is no way to get
rid of the vorticity without getting rid of the lift.
Winglets encourage the vortices to be shed at the
wingtips, not somewhere else along the span. This produces more
lift, since it is only the amount of span that carries circulation
that produces lift according to the Kutta-Zhukovsky theorem.
Still, as a general rule, adding a pair of six-foot-tall winglets
has no aerodynamic advantage compared to adding six feet of regular,
horizontal wing on each side.16
The bound vortex that produces the circulation that
supports the weight of the airplane should not be confused with
the little vortices produced by vortex generators (to re-energize
the boundary layer) as discussed in section 18.3.
3.12.2 Wake TurbulenceWhen air traffic control (ATC) tells you ``caution — wake
turbulence'' they are really telling you that some previous airplane
has left a wake vortex in your path. The wake
vortex from a large, heavy aircraft can easily flip a small aircraft
A heavy airplane like a C5-A flying slowly is the
biggest threat, because it needs lots of circulation to support
all that weight at a low airspeed. You would think that a C5-A
with flaps extended would be the absolute worst, but that is not
quite true. The flaps do increase the circulation-producing capability
of the wing, but they do not extend over the full span. Therefore
a part of the circulation is shed where the flaps end,
and another part is shed at the wingtips. If you fly into the
wake of another plane, two medium-strength vortices will cause
you less grief than a single full-strength vortex. Therefore,
you should expect that the threat from wake vortices is greatest
behind an airplane that is heavy, slow, and clean.
Like a common smoke ring, the wake vortex does not
just sit there, it moves. In this case it moves downward. A
common rule of thumb says they normally descend at about 500 feet
per minute, but the actual rate will depend on the wingspan and
coefficient of lift of the airplane that produced the vortex.
Vortices are part of the air; if the wind is blowing, the vortices
will be carried downwind. In fact, the reason wake vortices descend
is that the right vortex is carried downward by the flow field set up
by the left vortex, and the left vortex is carried downward by the
flow field set up by the right vortex. Superimposed on this flow
field, the overall wind blows the
vortices around in the obvious way.
When a vortex line gets close to the ground, it ``sees
its reflection''. That is, it moves as if there were being
acted on by a mirror-image vortex line a few feet below the ground.
This causes wake vortices to spread out — the left vortex starts
moving to the left, and the right vortex starts moving to the
* Avoiding Wake Turbulence ProblemsIf you are flying a light aircraft, avoid the airspace
below and behind a large aircraft. Avoiding the area for a minute
or two suffices, because a vortex that is older than that will
have lost enough intensity that it is probably not a serious problem.
If you are landing on the same runway as a preceding
large aircraft, you can avoid its wake vortices by flying a high,
steep approach, and landing at a point well beyond the point where
it landed. Remember, it doesn't produce vortices unless it is
producing lift. Assuming you are landing into the wind, the wind
can only help clear out the vortices for you.
If you are departing from the same runway as a preceding large
aircraft, you can avoid its vortices — in theory — if you leave
the runway at a point well before the point where it did, and if
you make sure that your climb-out profile stays above and/or behind
its. In practice, this might be hard to do, since the other aircraft
might be able to climb more steeply than you can. Also, since you are
presumably taking off into the wind, you need to worry that the
wind might blow the other plane's vortices toward you.
A light crosswind might keep a vortex on the runway
longer, by opposing its spreading motion. A less common problem is
that a crosswind might blow vortices from a parallel runway onto your
The technique that requires the least sophistication is to
delay your takeoff a few minutes, so the vortices can spread out and
be weakened by friction.
3.12.3 Induced DragHere are some more benefits of understanding circulation
and vortices: it explains induced drag, and explains why gliders have long skinny
wings. Induced drag is commonly said to be the ``cost''
of producing lift. But there is no law of physics that requires
a definite cost. If you could take a very large amount of air
and pull it downward very gently, you could support your weight
at very little cost. The cost you absolutely must pay is the
cost of making that trailing vortex. For every mile that the
airplane flies, each wingtip makes another mile of vortex. The
circulatory motion in that vortex involves nontrivial amounts
of kinetic energy, and that's why you have induced drag. A long
skinny wing will need less circulation than a short fat wing producing
the same lift. Gliders (which need to fly slowly with minimum
drag) therefore have very long skinny wings (limited only by strength;
it's hard to build something long, skinny, and strong).
3.12.4 Soft-Field TakeoffWe can now understand why
soft-field takeoff procedure works. When the aircraft is in
ground effect, it ``sees its reflection'' in the ground. If you are
flying 10 feet above the ground, the effect is the same as having a
mirror-image aircraft flying 10 feet below the ground. Its wingtip
vortices spin in the opposite direction and largely cancel
your wingtip vortices — greatly reducing induced
As discussed in section 13.4, in a soft-field takeoff, you
leave the ground at a very low airspeed, and then fly in ground effect
for a while. There will be no wheel friction (or damage) because the
wheels are not touching the ground. There will be very little induced
drag because of the ground effect, and
there will be very little parasite
drag because you are going slowly. The airplane
will accelerate like crazy. When you reach normal flying speed, you
raise the nose and fly away.
3.13 Frost on the WingsThe Federal Aviation Regulations prohibit takeoff
when there is frost adhering to the
wings or control surfaces, unless it is polished smooth.
It is interesting that they do not require it to be entirely removed,
just polished smooth.
There are very good aerodynamic reasons for this rule:
The most obvious effect of roughness on the wings is to create
a lot more drag, as seen in the right panel in figure 3.28,
which shows wind-tunnel data for a real airfoil (the NACA 631-412
airfoil; see reference 5). At cruise angle of attack, the drag
is approximately doubled; at higher angles of attack (corresponding to
lower airspeeds) it is even worse.
- The less obvious (yet more critical) problem is that
roughness causes the wing to stall at a considerably lower angle
of attack, lower coefficient of lift, and higher airspeed. This
can be seen in the left panel of figure 3.28. The pilot
of the frosty airplane could get a very nasty surprise.
As mentioned in section 3.4, Bernoulli's
principle cannot be trusted when energy is being removed from the
system by friction. Frost, by sticking up into the breeze, is very
effective in removing energy from the system. This tends to
de-energize the boundary layer, leading to separation which produces
It is interesting that at moderate and low angles
of attack (cruise airspeed and above) the frost has hardly any
effect on the coefficient of lift. This reinforces the point
made in section 3.11 that the velocity of the air right
at the surface, relative to the surface, is not what produces
Sometimes the air temperature is just above freezing, but due to
history or due to radiative cooling, the skin of the airplane is much
colder and covered with frost. A jug of warm water works wonders.
3.14 Consistent (Not Cumulative) Laws of PhysicsWe have seen that several physical principles are
involved in producing lift. Each of the following statements
is correct as far as it goes:
We now examine the relationship between these physical
principles. Do we get a little bit of lift because of Bernoulli,
and a little bit more because of Newton? No, the laws of physics
are not cumulative in this way.
The wing produces lift ``because'' it
is flying at an angle of attack.
- The wing produces lift ``because'' of
- The wing produces lift ``because'' of
- The wing produces lift ``because'' of Newton's law of action and
There is only one lift-producing process. Each of
the explanations itemized above concentrates on a different aspect
of this one process. The wing produces circulation in proportion
to its angle of attack (and its airspeed). This circulation means
the air above the wing is moving faster. This in turn produces
low pressure in accordance with Bernoulli's principle. The low
pressure pulls up on the wing and pulls down on the air in accordance
with all of Newton's laws.
3.15 Momentum in the AirFor an airplane in steady flight, the forces must balance. We know
from the Newton's third law18 that for every force there must be an equal and
opposite force somewhere, but the special idea here is that
there must be an equal and opposite force locally to maintain
The earth pulls down on the airplane (by gravity). This force is
balanced locally because the air pulls up on the airplane (by means of
pressure near the wings). Of course the same pressure that pulls up
on the airplane pulls down on the air; this force is transmitted from
one air parcel to another to another, all the way to the earth's
surface. At the earth's surface, pressure pushes up on the air and
pushes down on the earth. The downward force on the earth is just
enough to balance the fact that the airplane is pulling up on the
earth (by gravity).
Since force is just momentum per unit time, the same process can be
described by a big ``closed circuit'' of momentum flow. The earth
transfers downward momentum to the airplane (by gravity). The
airplane transfers downward momentum to the air (by pressure near the
wings). The momentum is then transferred from air parcel to air
parcel to air parcel. Finally the momentum is transferred back to the
earth (by pressure at the surface), completing the cycle.
You need to look at figure 3.27 to get the whole story. If
you look only at things like figure 3.2, you will never
understand how the momentum balance works, because that figure
doesn't tell the whole story. You might be tempted to make the
following erroneous argument:
To solve this paradox, remember that figure 3.2 does
not tell the whole story. It only shows the effects of the bound
vortex that runs along the wing, and does not
show the effects of the trailing vortices.
That is, it is only valid relatively close to the wing and relatively
far from the wingtips.
In figure 3.2, there is some upward momentum
ahead of the wing, and some downward momentum behind the wing.
- As the wing moves along, it carries the pattern of upwash and
downwash along with it.
- Therefore the total amount of upward and downward momentum in
the air is not changing as the wing moves along. No momentum is
being transferred to the air. Therefore no lift is being produced.
This is nonsense!
Look at that figure and choose a point, say, half a chord ahead of the
wing. You will see that the air has some upward
momentum at that point. All points above and below that point
within the frame of the figure also have upward momentum. But
it turns out that if you go up or down from that point more than a
wingspan or so, you will find that all the air has downward momentum.
This is caused by the trailing vortices, which induce a downward flow.
Near the wing the bound vortex dominates, but if you go higher or
lower the trailing vortices dominate.
In fact, if you add up all the momentum in an entire column of air,
for any column ahead of the wing, you will find that the total
vertical momentum is zero. The total effect of the trailing vortices
exactly cancels the total effect of the bound vortex.
If you consider points directly ahead of the wing (not above or
below), a slightly different sort of cancellation occurs. The effect
of the trailing vortices is never enough to actually reverse the flow;
there is always some upwash directly ahead of the wing, no
matter far ahead. But the effect of the trailing vortices greatly
reduces the magnitude, so the upwash pretty soon becomes negligible.
This is why it is reasonable to speak of ``undisturbed'' air ahead of
Behind the wing there is no cancellation of any kind; the
downwash of the wing is only reinforced by the downward flow
induced by the trailing vortices. There is plenty of downward
momentum in any air column behind the wing.
The general rule is simple: There is downward momentum in any air
column that passes through the vortex loop
(which is shown in figure 3.27). There is no momentum in any
air column that is ahead of the wing, outboard of the trailing
vortices, or aft of the starting vortex.
So now we can understand the momentum balance: As the airplane flies
along minute by minute, it imparts more and more downward momentum to
the air, by enlarging the region of downward-moving air behind it.
3.16 Summary: How a Wing Produces Lift
A wing is very effective at changing the speed
of the air. The air above is speeded up; the air below is slowed
down. Each air parcel gets a temporary change in speed and a
permanent offset in position.
- Bernoulli's principle asserts that a given parcel of air has
high velocity when it has low pressure, and vice versa.
- Below-atmospheric pressure above the wing is
much more pronounced than above-atmospheric pressure below the
- There is significant upwash ahead of the wing
and even more downwash behind the wing.
- The front stagnation line is well below and behind
the leading edge.
- The rear stagnation line is at or very near the
trailing edge. The Kutta condition says the air wants to flow
cleanly off the sharp trailing edge. This determines the amount
- An airfoil does not have to be curved
on top and/or flat on the bottom in order to work. A rounded
leading edge is a good idea, but even a barn door will fly.
- Air passing above and below the wing does not do so in
equal time. When lift is being produced, the air passes above the
wing in substantially less time.
- Lift is equal to circulation, times airspeed,
times density, times wingspan.
- Well below the stalling angle of attack, the
coefficient of lift is proportional to the angle of attack; the
circulation is proportional to the coefficient of lift times the
- Air is a fluid, not a bunch of bullets. The fluid has
pressure and velocity everywhere, not just where it meets the surface
of the wing.
- There is downward momentum in any air column behind the wing.
There is zero momentum in any air column ahead of the wing, outboard
of the trailing vortices, or aft of the starting vortex.
- Vortex lines cannot have loose ends; therefore
you cannot produce lift without producing wake vortices.
- Induced drag arises when you have low speed and/or short span,
because you are visiting a small amount of air and yanking it down
violently, producing strong wake vortices. In contrast there is very
little induced drag when you have high speed and/or long span,
because you are visiting a large amount of air, pulling it down
gently, producing weak wake vortices.
- These simulations are based on a
number of assumptions, including that the viscosity is small (but not
zero), the airspeed is small compared to the speed of sound, the
airflow is not significantly turbulent, no fluid can flow through
the surface of the wing, and the points of interest are close to
the wing and not too close to either wingtip.
- To be more precise: there is no wind in either of
the two dimensions that show up in figure 3.3.
There might be some flow in the third dimension (i.e. spanwise
along the stagnation line) but that isn't relevant to the present
- ... although for turbulent
flow, the stream lines can get so tangled that they lose any useful
discussed in section 2.2, I choose to measure angle of
attack in such a way that this zero-lift condition corresponds
to zero angle of attack, even for cambered wings.
- This was defined in section 2.12; see also
- By Bernoulli's principle, the
slowest air has the highest pressure. At the stagnation lines, the
air is stopped — which as slow as it can get! See section 3.4, especially figure 3.8.
course, if there were no atmospheric pressure below the wing, there
would be no way to have reduced pressure above the wing.
Fundamentally, atmospheric pressure below the wing is responsible for
supporting the weight of the airplane. The point is that pressure
changes above the wing are more pronounced than the pressure changes
below the wing.
- Newton's laws
are discussed in section 19.1.
- This is
a first-order equation, valid whenever the
pressure changes are a small percentage of the total atmospheric
pressure. See discussion below.
- ... but not always. See section 18.4 for a
airfoil designations aren't just serial numbers; the digits actually
contain information about the shape of the airfoil. For details
see reference 5.
- We are still
assuming negligible viscosity, small percentage pressure changes, no
turbulence in the fluid, no fluid flowing through the surface of the
wing, and a few other reasonable assumptions.
- Actually, you
never get 100% of the circulation predicted by the Kutta condition,
especially for crummy airfoils like barn doors. For nice airfoils
with a rounded leading edge, you get something like 99% of the
- Choose an
airplane where the stall warning indicator is
on the flapped section of the wing. This includes the Cessna C-152
and C-172, but not the C-182. It includes most Mooneys and the
Grumman Tiger, but
excludes Piper Cherokees and the Beech Bonanza.
assumes that the loop is big enough to include the places where
circulation is being produced (i.e. the wing and the boundary
the winglet solution may provide a practical advantage when taxiing
and parking. This is why Boeing put winglets (instead of additional
span) on the 747-400 — they wanted to be able to park in a standard
slot at the airport.
- Boundary layers, separation,
etc. are discussed in more detail in section 18.3.
section 19.1 for a discussion of the
laws of motion.
[Comments or questions]
Copyright © 1996-2001 jsd