Do recent
explanations solve the mysteries of aerodynamic lift?
NoOne Can Explain Why Planes Stay in the Air
NoOne Can Explain Why Planes Stay in the Air
Scientific
American, February 1, 2020
On a strictly mathematical level, engineers
know how to design planes that will stay aloft. But equations don't explain why
aerodynamic lift occurs.
There are two competing theories that
illuminate the forces and factors of lift. Both are incomplete explanations.
Aerodynamicists have recently tried to
close the gaps in understanding. Still, no consensus exists.
In December 2003, to commemorate the 100th
anniversary of the first flight of the Wright brothers, the New York Times ran
a story entitled “Staying Aloft; What Does Keep Them Up There?” The point of
the piece was a simple question: What keeps planes in the air? To answer it,
the Times turned to John D. Anderson, Jr., curator of aerodynamics at the
National Air and Space Museum and author of several textbooks in the field.
What Anderson said, however, is that there is
actually no agreement on what generates the aerodynamic force known as lift.
“There is no simple one-liner answer to this,” he told the Times. People give
different answers to the question, some with “religious fervor.” More than 15
years after that pronouncement, there are still different accounts of what
generates lift, each with its own substantial rank of zealous defenders. At
this point in the history of flight, this situation is slightly puzzling. After
all, the natural processes of evolution, working mindlessly, at random and without
any understanding of physics, solved the mechanical problem of aerodynamic lift
for soaring birds eons ago. Why should it be so hard for scientists to explain
what keeps birds, and airliners, up in the air?
Adding to the confusion is the fact that accounts
of lift exist on two separate levels of abstraction: the technical and the
nontechnical. They are complementary rather than contradictory, but they differ
in their aims. One exists as a strictly mathematical theory, a realm in which
the analysis medium consists of equations, symbols, computer simulations and
numbers. There is little, if any, serious disagreement as to what the
appropriate equations or their solutions are. The objective of technical
mathematical theory is to make accurate predictions and to project results that
are useful to aeronautical engineers engaged in the complex business of
designing aircraft.
But by themselves, equations are not
explanations, and neither are their solutions. There is a second, nontechnical
level of analysis that is intended to provide us with a physical, commonsense
explanation of lift. The objective of the nontechnical approach is to give us
an intuitive understanding of the actual forces and factors that are at work in
holding an airplane aloft. This approach exists not on the level of numbers and
equations but rather on the level of concepts and principles that are familiar
and intelligible to non-specialists.
It is on this second, nontechnical level where
the controversies lie. Two different theories are commonly proposed to explain
lift, and advocates on both sides argue their viewpoints in articles, in books
and online. The problem is that each of these two nontechnical theories is
correct in itself. But neither produces a complete explanation of lift, one
that provides a full accounting of all the basic forces, factors and physical
conditions governing aerodynamic lift, with no issues left dangling,
unexplained or unknown. Does such a theory even exist?
Two Competing Theories
By far the most popular explanation of lift is
Bernoulli’s theorem, a principle identified by Swiss mathematician Daniel
Bernoulli in his 1738 treatise, Hydrodynamica. Bernoulli came from a family of
mathematicians. His father, Johann, made contributions to the calculus, and his
Uncle Jakob coined the term “integral.” Many of Daniel Bernoulli’s
contributions had to do with fluid flow: Air is a fluid, and the theorem
associated with his name is commonly expressed in terms of fluid dynamics.
Stated simply, Bernoulli’s law says that the pressure of a fluid decreases as
its velocity increases, and vice versa.
Bernoulli’s theorem attempts to explain lift as
a consequence of the curved upper surface of an airfoil, the technical name for
an airplane wing. Because of this curvature, the idea goes, air traveling
across the top of the wing moves faster than the air moving along the wing’s
bottom surface, which is flat. Bernoulli’s theorem says that the increased
speed atop the wing is associated with a region of lower pressure there, which
is lift.
Mountains of empirical data from streamlines
(lines of smoke particles) in wind-tunnel tests, laboratory experiments on
nozzles and Venturi tubes, and so on provide overwhelming evidence that as
stated, Bernoulli’s principle is correct and true. Nevertheless, there are
several reasons that Bernoulli’s theorem does not by itself constitute a
complete explanation of lift. Although it is a fact of experience that air
moves faster across a curved surface, Bernoulli’s theorem alone does not
explain why this is so. In other words, the theorem does not say how the higher
velocity above the wing came about to begin with.
There are plenty of bad explanations for the
higher velocity. According to the most common one—the “equal transit time”
theory—parcels of air that separate at the wing’s leading edge must rejoin
simultaneously at the trailing edge. Because the top parcel travels farther
than the lower parcel in a given amount of time, it must go faster. The fallacy
here is that there is no physical reason that the two parcels must reach the
trailing edge simultaneously. And indeed, they do not: the empirical fact is
that the air atop moves much faster than the equal transit time theory could
account for.
There is also a notorious “demonstration” of
Bernoulli’s principle, one that is repeated in many popular accounts, YouTube
videos and even some textbooks. It involves holding a sheet of paper
horizontally at your mouth and blowing across the curved top of it. The page
rises, supposedly illustrating the Bernoulli effect. The opposite result ought
to occur when you blow across the bottom of the sheet: the velocity of the
moving air below it should pull the page downward. Instead, paradoxically, the
page rises.
The lifting of the curved paper when flow is
applied to one side “is not because air is moving at different speeds on the
two sides,” says Holger Babinsky, a professor of aerodynamics at the University
of Cambridge, in his article “How Do Wings Work?” To demonstrate this, blow
across a straight piece of paper—for example, one held so that it hangs down
vertically—and witness that the paper does not move one way or the other,
because “the pressure on both sides of the paper is the same, despite the
obvious difference in velocity.”
The second shortcoming of Bernoulli’s theorem
is that it does not say how or why the higher velocity atop the wing brings
lower pressure, rather than higher pressure, along with it. It might be natural
to think that when a wing’s curvature displaces air upward, that air is
compressed, resulting in increased pressure atop the wing. This kind of
“bottleneck” typically slows things down in ordinary life rather than speeding
them up. On a highway, when two or more lanes of traffic merge into one, the
cars involved do not go faster; there is instead a mass slowdown and possibly
even a traffic jam. Air molecules flowing atop a wing do not behave like that,
but Bernoulli’s theorem does not say why not.
The third problem provides the most decisive
argument against regarding Bernoulli’s theorem as a complete account of lift:
An airplane with a curved upper surface is capable of flying inverted. In
inverted flight, the curved wing surface becomes the bottom surface, and
according to Bernoulli’s theorem, it then generates reduced pressure below the
wing. That lower pressure, added to the force of gravity, should have the
overall effect of pulling the plane downward rather than holding it up.
Moreover, aircraft with symmetrical airfoils, with equal curvature on the top
and bottom—or even with flat top and bottom surfaces—are also capable of flying
inverted, so long as the airfoil meets the oncoming wind at an appropriate
angle of attack. This means that Bernoulli’s theorem alone is insufficient to
explain these facts.
The other theory of lift is based on Newton’s
third law of motion, the principle of action and reaction. The theory states
that a wing keeps an airplane up by pushing the air down. Air has mass, and
from Newton’s third law it follows that the wing’s downward push results in an
equal and opposite push back upward, which is lift. The Newtonian account
applies to wings of any shape, curved or flat, symmetrical or not. It holds for
aircraft flying inverted or right-side up. The forces at work are also familiar
from ordinary experience—for example, when you stick your hand out of a moving
car and tilt it upward, the air is deflected downward, and your hand rises. For
these reasons, Newton’s third law is a more universal and comprehensive
explanation of lift than Bernoulli’s theorem.
But taken by itself, the principle of action
and reaction also fails to explain the lower pressure atop the wing, which
exists in that region irrespective of whether the airfoil is cambered. It is
only when an airplane lands and comes to a halt that the region of lower pressure
atop the wing disappears, returns to ambient pressure, and becomes the same at
both top and bottom. But as long as a plane is flying, that region of lower
pressure is an inescapable element of aerodynamic lift, and it must be
explained.
Historical Understanding
Neither Bernoulli nor Newton was consciously
trying to explain what holds aircraft up, of course, because they lived long
before the actual development of mechanical flight. Their respective laws and
theories were merely repurposed once the Wright brothers flew, making it a
serious and pressing business for scientists to understand aerodynamic lift.
Most of these theoretical accounts came from
Europe. In the early years of the 20th century, several British scientists
advanced technical, mathematical accounts of lift that treated air as a perfect
fluid, meaning that it was incompressible and had zero viscosity. These were
unrealistic assumptions but perhaps understandable ones for scientists faced
with the new phenomenon of controlled, powered mechanical flight. These
assumptions also made the underlying mathematics simpler and more
straightforward than they otherwise would have been, but that simplicity came
at a price: however successful the accounts of airfoils moving in ideal gases
might be mathematically, they remained defective empirically.
In Germany, one of the scientists who applied
themselves to the problem of lift was none other than Albert Einstein. In 1916
Einstein published a short piece in the journal Die Naturwissenschaften
entitled “Elementary Theory of Water Waves and of Flight,” which sought to
explain what accounted for the carrying capacity of the wings of flying
machines and soaring birds. “There is a lot of obscurity surrounding these
questions,” Einstein wrote. “Indeed, I must confess that I have never
encountered a simple answer to them even in the specialist literature.”
Einstein then proceeded to give an explanation
that assumed an incompressible, frictionless fluid—that is, an ideal fluid.
Without mentioning Bernoulli by name, he gave an account that is consistent
with Bernoulli’s principle by saying that fluid pressure is greater where its
velocity is slower, and vice versa. To take advantage of these pressure
differences, Einstein proposed an airfoil with a bulge on top such that the
shape would increase airflow velocity above the bulge and thus decrease
pressure there as well.
Einstein probably thought that his ideal-fluid
analysis would apply equally well to real-world fluid flows. In 1917, on the
basis of his theory, Einstein designed an airfoil that later came to be known
as a cat’s-back wing because of its resemblance to the humped back of a
stretching cat. He brought the design to aircraft manufacturer LVG
(Luftverkehrsgesellschaft) in Berlin, which built a new flying machine around
it. A test pilot reported that the craft waddled around in the air like “a
pregnant duck.” Much later, in 1954, Einstein himself called his excursion into
aeronautics a “youthful folly.” The individual who gave us radically new
theories that penetrated both the smallest and the largest components of the
universe nonetheless failed to make a positive contribution to the
understanding of lift or to come up with a practical airfoil design.
Toward a Complete Theory of Lift
Contemporary scientific approaches to aircraft
design are the province of computational fluid dynamics (CFD) simulations and
the so-called Navier-Stokes equations, which take full account of the actual
viscosity of real air. The solutions of those equations and the output of the
CFD simulations yield pressure-distribution predictions, airflow patterns and
quantitative results that are the basis for today’s highly advanced aircraft
designs. Still, they do not by themselves give a physical, qualitative
explanation of lift.
In recent years, however, leading
aerodynamicist Doug McLean has attempted to go beyond sheer mathematical
formalism and come to grips with the physical cause-and-effect relations that
account for lift in all of its real-life manifestations. McLean, who spent most
of his professional career as an engineer at Boeing Commercial Airplanes, where
he specialized in CFD code development, published his new ideas in the 2012
text Understanding Aerodynamics: Arguing from the Real Physics.
Considering that the book runs to more than 500
pages of fairly dense technical analysis, it is surprising to see that it
includes a section (7.3.3) entitled “A Basic Explanation of Lift on an Airfoil,
Accessible to a Nontechnical Audience.” Producing these 16 pages was not easy
for McLean, a master of the subject; indeed, it was “probably the hardest part
of the book to write,” the author says. “It saw more revisions than I can
count. I was never entirely happy with it.”
McLean’s complex explanation of lift starts
with the basic assumption of all ordinary aerodynamics: the air around a wing
acts as “a continuous material that deforms to follow the contours of the
airfoil.” That deformation exists in the form of a deep swath of fluid flow
both above and below the wing. “The airfoil affects the pressure over a wide
area in what is called a pressure field,” McLean writes. “When lift is
produced, a diffuse cloud of low pressure always forms above the airfoil, and a
diffuse cloud of high pressure usually forms below. Where these clouds touch the
airfoil they constitute the pressure difference that exerts lift on the
airfoil.”
The wing pushes the air down, resulting in a
downward turn of the airflow. The air above the wing is sped up in accordance
with Bernoulli’s principle. In addition, there is an area of high pressure
below the wing and a region of low pressure above. This means that there are
four necessary components in McLean’s explanation of lift: a downward turning
of the airflow, an increase in the airflow’s speed, an area of low pressure and
an area of high pressure.
But it is the interrelation among these four
elements that is the most novel and distinctive aspect of McLean’s account.
“They support each other in a reciprocal cause-and-effect relationship, and
none would exist without the others,” he writes. “The pressure differences
exert the lift force on the airfoil, while the downward turning of the flow and
the changes in flow speed sustain the pressure differences.” It is this
interrelation that constitutes a fifth element of McLean’s explanation: the
reciprocity among the other four. It is as if those four components
collectively bring themselves into existence, and sustain themselves, by
simultaneous acts of mutual creation and causation.
There seems to be a hint of magic in this synergy.
The process that McLean describes seems akin to four active agents pulling up
on one another’s bootstraps to keep themselves in the air collectively. Or, as
he acknowledges, it is a case of “circular cause-and-effect.” How is it
possible for each element of the interaction to sustain and reinforce all of
the others? And what causes this mutual, reciprocal, dynamic interaction?
McLean’s answer: Newton’s second law of motion.
Newton’s second law states that the
acceleration of a body, or a parcel of fluid, is proportional to the force
exerted on it. “Newton’s second law tells us that when a pressure difference
imposes a net force on a fluid parcel, it must cause a change in the speed or
direction (or both) of the parcel’s motion,” McLean explains. But reciprocally,
the pressure difference depends on and exists because of the parcel’s
acceleration.
Aren’t we getting something for nothing here?
McLean says no: If the wing were at rest, no part of this cluster of mutually
reinforcing activity would exist. But the fact that the wing is moving through
the air, with each parcel affecting all of the others, brings these
co-dependent elements into existence and sustains them throughout the flight.
Turning on the Reciprocity of Lift
Soon after the publication of Understanding
Aerodynamics, McLean realized that he had not fully accounted for all the
elements of aerodynamic lift, because he did not explain convincingly what
causes the pressures on the wing to change from ambient. So, in November 2018,
McLean published a two-part article in The Physics Teacher in which he proposed
“a comprehensive physical explanation” of aerodynamic lift.
Although the article largely restates McLean’s
earlier line of argument, it also attempts to add a better explanation of what
causes the pressure field to be nonuniform and to assume the physical shape
that it does. In particular, his new argument introduces a mutual interaction
at the flow field level so that the nonuniform pressure field is a result of an
applied force, the downward force exerted on the air by the airfoil.
Whether McLean’s section 7.3.3 and his
follow-up article are successful in providing a complete and correct account of
lift is open to interpretation and debate. There are reasons that it is
difficult to produce a clear, simple and satisfactory account of aerodynamic
lift. For one thing, fluid flows are more complex and harder to understand than
the motions of solid objects, especially fluid flows that separate at the
wing’s leading edge and are subject to different physical forces along the top
and bottom. Some of the disputes regarding lift involve not the facts
themselves but rather how those facts are to be interpreted, which may involve
issues that are impossible to decide by experiment.
Nevertheless, there are at this point only a
few outstanding matters that require explanation. Lift, as you will recall, is
the result of the pressure differences between the top and bottom parts of an
airfoil. We already have an acceptable explanation for what happens at the bottom
part of an airfoil: the oncoming air pushes on the wing both vertically
(producing lift) and horizontally (producing drag). The upward push exists in
the form of higher pressure below the wing, and this higher pressure is a
result of simple Newtonian action and reaction.
Things are quite different at the top of the
wing, however. A region of lower pressure exists there that is also part of the
aerodynamic lifting force. But if neither Bernoulli’s principle nor Newton’s
third law explains it, what does? We know from streamlines that the air above
the wing adheres closely to the downward curvature of the airfoil. But why must
the parcels of air moving across the wing’s top surface follow its downward
curvature? Why can’t they separate from it and fly straight back?
Mark Drela, a professor of fluid dynamics at
the Massachusetts Institute of Technology and author of Flight Vehicle
Aerodynamics, offers an answer: “If the parcels momentarily flew off tangent to
the airfoil top surface, there would literally be a vacuum created below them,”
he explains. “This vacuum would then suck down the parcels until they mostly
fill in the vacuum, i.e., until they move tangent to the airfoil again. This is
the physical mechanism which forces the parcels to move along the airfoil
shape. A slight partial vacuum remains to maintain the parcels in a curved
path.”
This drawing away or pulling down of those air
parcels from their neighboring parcels above is what creates the area of lower
pressure atop the wing. But another effect also accompanies this action: the
higher airflow speed atop the wing. “The reduced pressure over a lifting wing
also ‘pulls horizontally’ on air parcels as they approach from upstream, so
they have a higher speed by the time they arrive above the wing,” Drela says.
“So the increased speed above the lifting wing can be viewed as a side effect
of the reduced pressure there.”
But as always, when it comes to explaining lift
on a nontechnical level, another expert will have another answer. Cambridge
aerodynamicist Babinsky says, “I hate to disagree with my esteemed colleague
Mark Drela, but if the creation of a vacuum were the explanation, then it is
hard to explain why sometimes the flow does nonetheless separate from the
surface. But he is correct in everything else. The problem is that there is no
quick and easy explanation.”
Drela himself concedes that his explanation is
unsatisfactory in some ways. “One apparent problem is that there is no
explanation that will be universally accepted,” he says. So where does that
leave us? In effect, right where we started: with John D. Anderson, who stated,
“There is no simple one-liner answer to this.”