A
fluid flowing past the surface of a body
exerts a
force on it.
Lift is
defined to be the
component of this force
that is
perpendicular to the
oncoming flow direction. It contrasts with the
drag force, which is defined to be
the component of the fluid-dynamic force
parallel to the flow direction.
Overview
If the fluid is air, the force is called an
aerodynamic force. An
airfoil is a streamlined shape that is capable of
generating significantly more lift than drag. Aerodynamic lift is
commonly associated with the
wing of a
fixed-wing aircraft, although lift is
also generated by
propellers;
helicopter rotors;
rudders,
sails and
keels on
sailboats;
hydrofoils;
wings
on
auto racing cars;
wind turbines and other streamlined objects.
While common meanings of the word "
lift" suggest that lift opposes gravity,
lift can be in any direction. When an aircraft is flying straight
and level (
cruise) most of the lift
opposes gravity. However, when an aircraft is
climbing,
descending, or
banking in a turn, for example, the
lift is tilted with respect to the vertical. Lift may also be
entirely downwards in some
aerobatic
manoeuvres, or on the wing on a racing car. In this last case,
the term
downforce is often used. Lift may
also be horizontal, for instance on a
sail on a
sailboat
Non-streamlined objects such as bluff bodies and plates (not
parallel to the flow) may also generate lift when moving relative
to the fluid. This lift may be steady, or it may
oscillate due to
vortex shedding. Interaction of the object's
flexibility with the vortex shedding may enhance the effects of
fluctuating lift and cause
vortex-induced vibrations.
Description of lift on an airfoil
There are several ways to explain how an airfoil generates lift.
Some are more complicated or more mathematically rigorous than
others; some have been shown to be incorrect. This article will
start with the simpler, more common explanations.
Newton's laws: Lift and the deflection of the flow
Deflection
One way to explain the phenomenon of lift is to observe the direct
relationship between the lift force and the downward deflection of
the air by the wing.
John D.
Anderson explains it as follows:
"
...the wing exerts a force on the air, pushing the flow
downward. From Newton's third law, the equal and opposite
reaction produces a lift."
This explanation relies on the second and third of
Newton's laws of motion:
The net
force on an object is equal to its rate of momentum change and
To every action there
is an equal and opposite reaction.
This explanation of lift was used in the 1944 book
Stick and Rudder by aviation writer
Wolfgang Langewiesche: "…
the wing keeps the airplane up by pushing the air down."
"The main fact of all heaver-than-air flight is this:
the wing
keeps the airplane up by pushing the air down." In:
Consequently, the lift is directly related to the deflection of the
flow field behind the airfoil.
Flow turning
Airstreams around an airfoil in a wind
tunnel.
Another way to think about it is to observe that the air "turns" as
it passes the airfoil and follows a path that is curved. When
airflow changes direction, a force is generated.
If one wants to think in terms of air pressure,
pressure is just force per unit area so wherever
there is force there is pressure. Curved streamlines imply a force
and therefore also pressure. "
...where there is curved fluid
flow, there is a pressure difference (i.e., lift) "
This pressure difference implies higher pressure on the underside
of the wing and lower pressure on the upper side.
Criticisms of deflection/turning
- While the deflection theory correctly reasons that deflection
implies that there must be a force on the airfoil, it does not
explain why the air is deflected.
- Intuitively, one can say that the air follows the curve of the
foil ("Most students will be happy with the streamline pattern
around a lifting wing ...because it intuitively looks right"),
but this is not very rigorous or precise.
- The theory, while correct in as far as it goes, is not
sufficiently rigorous or precise to allow one to do
engineering.
- Thus, most textbooks on aerodynamics use a more complex model
to explain lift.
"Popular" explanation based on equal transit-time
An illustration of the equal
transit-time theory.
An explanation of lift frequently encountered in basic or popular
sources is the equal transit-time theory. Equal
transit-time states that because of the longer path of the upper
surface of an airfoil, the air going over the top must go faster in
order to catch up with the air flowing around the bottom.
i.e. the parcels of air that are divided at the leading edge and
travel above and below an airfoil must rejoin when they reach the
trailing edge. Bernoulli's
Principle is then cited to conclude that since the air moves
faster on the top of the wing the air pressure must be lower. This
pressure difference pushes the wing up.
However, equal transit time is not accurate and the fact that this
is not generally the case can be readily observed. Although it is
true that the air moving over the top of a wing generating lift
does move faster, there is no requirement for equal transit time.
In fact the air moving over the top of an airfoil generating lift
is always moving much faster than the equal transit theory would
imply.
The assertion that the air must arrive simultaneously at the
trailing edge is sometimes referred to as the "Equal Transit-Time
Fallacy".
Note that while this theory depends on Bernoulli's Principle, the
fact that this theory has been discredited does not imply that
Bernoulli's Principle is incorrect.
A more rigorous physical description
Lift is generated in accordance with the fundamental principles of
physics. The most relevant physics reduce to
three principles:
In the last principle, the pressure depends on the other flow
properties, such as its mass density,
through the (thermodynamic) equation of state, while the shear
stresses are related to the flow through the air's viscosity. Application of the viscous shear
stresses to Newton's second law for an airflow results in the
Navier–Stokes
equations. But in many instances approximations suffice for a
good description of lifting airfoils: in large parts of the flow
viscosity may be neglected. Such an inviscid flow can be described mathematically
through the Euler
equations, resulting from the Navier-Stokes equations when the
viscosity is neglected.
The Euler equations for a steady and
inviscid flow can be integrated along a streamline,
resulting in Bernoulli's
equation. The particular form of Bernoulli's equation found
depends on the equation of state
used. At low Mach numbers,
compressibility effects may be neglected, resulting in an incompressible flow approximation. In
incompressible and inviscid flow the Bernoulli equation is just an
integration of Newton's second law—in the form of the description
of momentum evolution by the Euler
equations—along a streamline.
Explaining lift while considering all of the principles involved is
a complex task and is not easily simplified. As a result, there are
numerous different explanations of lift with different levels of
rigour and complexity. For example, there is an explanation based
directly on Newton’s laws of motion; and an explanation based on
Bernoulli’s principle. Neither of these explanations is incorrect,
but each appeals to a different audience.
In order to explain lift as it applies to an airplane wing,
consider the incompressible flow around a 2-D, symmetric airfoil at positive angle
of attack in a uniform free stream. Instead of considering the
case where an airfoil moves through a fluid as seen by a stationary
observer, it is equivalent and simpler to consider the picture when
the observer follows the airfoil and the fluid moves past it.
Lift in an established flow
Streamlines around a NACA 0012 airfoil
at moderate angle of attack.
If one takes the experimentally observed flow around an airfoil as
a starting point, then lift can be explained in terms of pressures using Bernoulli's principle (which can be
derived from Newton's second
law) and conservation of mass.
The image to the right shows the streamlines over a NACA
0012 airfoil computed using potential
flow theory, a simplified model of the real flow. The flow
approaching an airfoil can be divided into two
streamtubes, which are defined based on the area between
two streamlines. By definition, fluid never crosses a streamline in
a steady flow; hence mass is conserved
within each streamtube. One streamtube travels over the upper
surface, while the other travels over the lower surface; dividing
these two tubes is a dividing line (the stagnation streamline) that
intersects the airfoil on the lower surface, typically near to the
leading edge. The stagnation streamline leaves the airfoil at the
sharp trailing edge, a feature of the flow known as the Kutta condition. In calculating the flow
shown, the Kutta condition was imposed as an initial assumption;
the justification for this assumption is explained below.
The upper stream tube constricts as it flows up and around the
airfoil, a part of the so-called upwash. From
the conservation of mass, the flow speed must increase as the
stream tube area decreases. The area of the lower stream tube
increases, causing the flow inside the tube to slow down. It is
typically the case that the air parcels
traveling over the upper surface will reach the trailing edge
before those traveling over the bottom.
From Bernoulli's principle, the pressure on the upper surface where
the flow is moving faster is lower than the pressure on the lower
surface. The pressure difference thus creates a net aerodynamic force, pointing upward and
downstream to the flow direction. The component of the force normal
to the free stream is considered to be lift; the component parallel
to the free stream is drag. In
conjunction with this force by the air on the airfoil, by Newton's third law, the airfoil imparts
an equal-and-opposite force on the surrounding air that creates the
downwash. Measuring the momentum
transferred to the downwash is another way to determine the amount
of lift on the airfoil. , pp. 68–69 and pp. 153–155.
Flowfield formation
In attempting to explain why the flow follows the upper surface of
the airfoil, the situation gets considerably more complex. It is
here that many simplifications are made in presenting lift to
various audiences, some of which are explained
after this section.
Consider the case of an airfoil accelerating from rest in a
viscous flow. Lift depends entirely on the
nature of viscous flow past certain bodies: in inviscid flow (i.e. assuming that viscous
forces are negligible in comparison to inertial forces), there is
no lift without imposing a net circulation, the proper amount
of which can be determined by applying the Kutta condition. In a
viscous flow like in the physical world, however, the lift and
other properties arise naturally as described here.
When there is no flow, there is no lift and the forces acting on
the airfoil are zero. At the instant when the flow is “turned on”,
the flow is undeflected downstream of the
airfoil and there are two stagnation
points on the airfoil (where the flow velocity is zero): one
near the leading edge on the bottom surface, and another on the
upper surface near the trailing edge. The dividing line between the
upper and lower streamtubes mentioned above intersects the body at
the stagnation points. Since the flow speed is zero at these
points, by Bernoulli's principle the static pressure at these points is at a
maximum. As long as the second stagnation point is at its initial
location on the upper surface of the wing, the circulation around the airfoil
is zero and, in accordance with the Kutta–Joukowski theorem,
there is no lift. The net pressure difference between the upper and
lower surfaces is zero.
The effects of viscosity are contained within a thin layer of fluid
called the boundary layer, close to
the body. As flow over the airfoil commences, the flow along the
lower surface turns at the sharp trailing edge and flows along the
upper surface towards the upper stagnation point. The flow in the
vicinity of the sharp trailing edge is very fast and the resulting
viscous forces cause the boundary layer to accumulate into a vortex
on the upper side of the airfoil between the trailing edge and the
upper stagnation point. This is called the starting vortex. The starting vortex and the
bound vortex around the surface of the wing are two halves of a
closed loop. As the starting vortex increases in strength the bound
vortex also strengthens, causing the flow over the upper surface of
the airfoil to accelerate and drive the upper stagnation point
towards the sharp trailing edge. As this happens, the starting vortex is shed into the wake, and
is a necessary condition to produce lift on an airfoil. If the flow
were stopped, there would be a corresponding "stopping vortex".
Despite being an idealization of the real world, the “vortex
system” set up around a wing is both real and observable; the
trailing vortex sheet most noticeably rolls up into wing-tip vortices.
The upper stagnation point continues moving downstream until it is
coincident with the sharp trailing edge (as stated by the Kutta
condition). The flow downstream of the airfoil is deflected
downward from the free-stream direction and, from the reasoning
above in the basic explanation, there is now a net pressure
difference between the upper and lower surfaces and an aerodynamic
force is generated.
Other alternative explanations for the generation of lift
Many other alternative explanations for the generation of lift by
an airfoil have been put forward, of which a few are presented
here. Most of them are intended to explain the phenomenon of lift
to a general audience. Although the explanations may share features
in common with the explanation above, additional assumptions and
simplifications may be introduced. This can reduce the validity of
an alternative explanation to a limited sub-class of lift
generating conditions, or might not allow a quantitative analysis.
Several theories introduce assumptions which proved to be wrong,
like the equal transit-time theory.
Coandă effect
In a limited sense, the Coandă effect refers to the
tendency of a fluid jet to stay attached to an adjacent surface
that curves away from the flow, and the resultant entrainment of ambient air into
the flow. The effect is named for Henri Coandă, the Romanian
aerodynamicist who exploited it in many of his
patents.
One of the first known uses was in his patent for a high-lift
device that used a fan of gas exiting at high speed from an
internal compressor. This circular spray was directed radially over
the top of a curved surface shaped like a lens to decrease the
pressure on that surface. The total lift for the device was caused
by the difference between this pressure and that on the bottom of
the craft. Two aircraft, the Antonov An-72
and An-74 "Coaler", use the exhaust from top-mounted jet
engines flowing over the wing to enhance lift, as do the Boeing YC-14 and the McDonnell Douglas YC-15.
The effect is also used in high-lift devices such as a blown flap.
More broadly, some consider the effect to include the tendency of
any fluid boundary layer to adhere to
a curved surface, not just the boundary layer accompanying a fluid
jet. It is in this broader sense that the Coandă effect is used by
some to explain lift. Jef Raskin, for
example, describes a simple demonstration, using a straw to blow
over the upper surface of a wing. The wing deflects upwards, thus
supposedly demonstrating that the Coandă effect creates lift. This
demonstration correctly demonstrates the Coandă effect as a fluid
jet (the exhaust from a straw) adhering to a curved surface (the
wing). However, the upper surface in this flow is a complicated,
vortex-laden mixing layer, while on the lower surface the flow is
quiescent. The physics of this demonstration are very different
from that of the general flow over the wing. The usage in this
sense is encountered in some popular references on aerodynamics. In
the aerodynamics field, the Coandă effect is commonly defined in
the more limited sense above and viscosity
is used to explain why the boundary layer attaches to the surface
of a wing.
In terms of a difference in areas
When a fluid flows relative to a solid body, the body obstructs the
flow, causing some of the fluid to change its speed and direction
in order to flow around the body. The obstructive nature of the
solid body causes the streamlines to move closer
together in some places, and further apart in others.When fluid
flows past a 2-D cambered
airfoil at zero angle of attack, the upper surface has a
greater area (that is, the interior area of the airfoil above the
chordline) than the lower surface
and hence presents a greater obstruction to the fluid than the
lower surface. This asymmetry causes the streamlines in the fluid
flowing over the upper surface to move closer together than the
streamlines over the lower surface. As a consequence of mass
conservation, the reduced area between the streamlines over the
upper surface results in a higher velocity than that over the lower
surface. The upper streamtube is squashed the most in the nose
region ahead of the maximum thickness of the airfoil, causing the
maximum velocity to occur ahead of the maximum thickness.
In accordance with Bernoulli's
principle, where the fluid is moving faster the pressure is
lower, and where the fluid is moving slower the pressure is
greater. The fluid is moving faster over the upper surface,
particularly near the leading edge, than over the lower surface so
the pressure on the upper surface is lower than the pressure on the
lower surface. The difference in pressure between the upper and
lower surfaces results in lift.
Methods to determine lift on an airfoil
Lift coefficient
If the lift coefficient for a wing at a specified angle of attack
is known (or estimated using a method such as thin-airfoil theory), then the
lift produced for specific flow conditions can be determined using
the following equation:
L = \tfrac12\rho v^2 A C_L
where
This equation is basically the same as the drag equation, only the lift/drag coefficient
is different.
Kutta–Joukowski theorem
Lift can be calculated using potential
flow theory by imposing a circulation. It is often used
by practicing aerodynamicists as a convenient quantity in
calculations, for example thin-airfoil theory and lifting-line theory.
The circulation \Gamma is the line
integral of the velocity of the air, in a closed loop around
the boundary of an airfoil. It can be understood as the total
amount of "spinning" (or vorticity) of air
around the airfoil. The section lift/span L' can be calculated
using the Kutta–Joukowski
theorem:
- L' = -\rho V \Gamma\,
where \rho is the air density, V is the free-stream airspeed.
Kelvin's circulation
theorem states that circulation is conserved. There is
conservation of the air's angular momentum. When an aircraft is at
rest, there is no circulation.
The challenge when using the Kutta–Joukowski theorem to determine
lift is to determine the appropriate circulation for a particular
airfoil. In practice, this is done by applying the Kutta condition, which uniquely prescribes
the circulation for a given geometry and free-stream
velocity.
A physical understanding of the theorem can be observed in the
Magnus effect, which is a lift force
generated by a spinning cylinder in a free stream. Here the
necessary circulation is induced by the mechanical rotation acting
on the boundary layer, causing it to induce a faster flow around
one side of the cylinder and a slower flow around the other. The
asymmetric distribution of airspeed around the cylinder then
produces a circulation in the outer inviscid flow.
Pressure integration
The force on the wing can be examined in terms of the pressure differences above and below the wing,
which can be related to velocity changes by Bernoulli's principle.
The total lift force is the integral of
vertical pressure forces over the entire wetted surface area of the
wing:
- L = \oint p\mathbf{n} \cdot\mathbf{k} \; \mathrm{d}A,
where:
- L is the lift,
- A is the wing surface area
- p is the value of the pressure,
- n is the normal unit vector pointing into the
wing, and
- k is the vertical unit vector, normal to the
freestream direction.
The above lift equation neglects the skin
friction forces, which typically have a negligible contribution
to the lift compared to the pressure forces. By using the
streamwise vector i parallel to the freestream in
place of k in the integral, we obtain an
expression for the pressure drag
D_{p} (which includes induced drag in a 3D wing). If we use the
spanwise vector j, we obtain the side force
Y.
\begin{align}
D_p &= \oint p\mathbf{n} \cdot\mathbf{i} \; \mathrm{d}A,
\\[1.2ex]
Y &= \oint p\mathbf{n} \cdot\mathbf{j} \; \mathrm{d}A.
\end{align}
One method for calculating the pressure is Bernoulli's equation, which is the
mathematical expression of Bernoulli's principle. This method
ignores the effects of viscosity, which
can be important in the boundary
layer and to predict friction
drag, which is the other component of the total drag in addition to
D_{p}.
The Bernoulli principle states that the sum total of energy within
a parcel of fluid remains constant as long as no energy is added or
removed. It is a statement of the principle of the conservation of
energy applied to flowing fluids.
A substantial simplification of this proposes that as other forms
of energy changes are inconsequential during the flow of air around
a wing and that energy transfer in/out of the air is not
significant, then the sum of pressure energy and speed energy for
any particular parcel of air must be constant. Consequently, an
increase in speed must be accompanied by a decrease in pressure and
vice-versa. It should be noted that this is not a causational
relationship. Rather, it is a coincidental relationship, whatever
causes one must also cause the other as energy can neither be
created nor destroyed. It is named for the Dutch-Swiss mathematician and
scientist Daniel Bernoulli, though it was previously
understood by Leonhard Euler and
others.
Bernoulli's principle provides an explanation of pressure
difference in the absence of air density and temperature variation
(a common approximation for low-speed aircraft). If the air density
and temperature are the same above and below a wing, a naive
application of the ideal gas law
requires that the pressure also be the same. Bernoulli's principle,
by including air velocity, explains this pressure difference. The
principle does not, however, specify the air velocity. This must
come from another source, e.g., experimental data.
In order to solve for the velocity of inviscid flow around a wing,
the Kutta condition must be applied
to simulate the effects of inertia and viscosity. The Kutta
condition allows for the correct choice among an infinite number of
flow solutions that otherwise obey the laws of conservation of mass and conservation of momentum.
Lift forces on bluff bodies
The flow around bluff bodies may also generate lift, besides a
strong drag force. For instance, the flow around a circular
cylinder generates a Kármán vortex street:
vortices being shed in an alternating fashion
from each side of the cylinder. The oscillatory nature of the flow
is reflected in the fluctuating lift force on the cylinder, whereas
the mean lift force is negligible. The lift
force frequency is characterised by the
dimensionless Strouhal number, which depends (among
others) on the Reynolds number of
the flow.
For a flexible structure, this oscillatory lift force may induce
vortex-induced vibrations.
Under certain conditions — for instance resonance or strong spanwise correlation of the lift force — the resulting
motion of the structure due to the lift fluctuations may be
strongly enhanced. Such vibrations may pose problems, even
collapse, in man-made tall structures like for instance industrial
chimneys, if not properly taken care of in
the design.
See also
References and notes
- Clancy, L.J., Aerodynamics, Section 5.2
- The amount of lift will be (usually slightly) more or
less than gravity depending on the thrust level and vertical
alignment of the thrust line. A side thrust line will result in
some lift opposing side thrust as well.
- Clancy, L.J., Aerodynamics, Section 14.6
- "The wing deflects the airflow such that the mean velocity
vector
behind the wing is canted slightly downward (…). Hence, the wing
imparts a downward component of momentum to the air; that is, the
wing exerts a force on the air, pushing the flow downward. From
Newton's third law, the equal and opposite reaction produces a
lift."
- , Vol. 1, §10–1 and §10–2.
- "...if a streamline is curved, there must be a pressure
gradient across the streamline..."
- “This is simply not true.” Anderson, John D. Jr,
Introduction to Flight, p.355 (5th edition), McGraw-Hill
ISBN 0-07-282569-3
- A visualization of the typical retarded flow over the lower
surface of the wing and the accelerated flow over the upper surface
starts at 5:29 in the video.
- A false explanation for lift has been put forward in mainstream
books, and even in scientific exhibitions. Known as the "equal
transit-time" explanation, it states that the parcels of air which
are divided by an airfoil must rejoin again; because of the greater
curvature (and hence longer path) of the upper surface of an
aerofoil, the air going over the top must go faster in order to
"catch up" with the air flowing around the bottom. Therefore,
because of its higher speed the pressure of the air above the
airfoil must be lower. Despite the fact that this "explanation" is
probably the most common of all, it is false. It has recently been
dubbed the "Equal transit-time fallacy."
- ...it leaves the impression that Professor Bernoulli is somehow
to blame for the "equal transit time" fallacy...
- The fallacy of equal transit time can be deduced from
consideration of a flat plate, which will indeed produce lift, as
anyone who has handled a sheet of plywood in the wind can
testify.
- Fallacy 1: Air takes the same time to move across the top of an
aerofoil as across the bottom.
- Clancy, L.J., Aerodynamics, Figure 4.7
- Clancy, L.J., Aerodynamics, Figure 4.8
- Clancy, L.J., Aerodynamics, Section 7.27
- Clancy, L.J., Aerodynamics, Sections 4.5 and 4.6
- Anderson, John D. (2004), Introduction to Flight,
Section 5.7 (5th edition), McGraw-Hill. ISBN 0-07-282569-3
Further reading
- Introduction to Flight, John D. Anderson, Jr.,
McGraw-Hill, ISBN 0-07-299071-6 — The author is the Curator of
Aerodynamics at the Smithsonian Institution's National Air &
Space Museum and Professor Emeritus at the University of
Maryland.
- Understanding Flight, by David Anderson and Scott
Eberhardt, McGraw-Hill, ISBN 0-07-136377-7 — The authors are a
physicist and an aeronautical engineer. They explain flight in
non-technical terms and specifically address the equal-transit-time
myth. Turning of the flow around the wing is attributed to the
Coanda effect, which is quite controversial.
- Aerodynamics, Clancy, L.J. (1975), Section 4.8, Pitman
Publishing Limited, London ISBN 0 273 01120 0.
- Quest for an improved explanation of lift Jaako
Hoffren (Helsinki Univ. of Technology, Espoo, Finland)
AIAA-2001-872 Aerospace Sciences Meeting and Exhibit, 39th, Reno,
NV, Jan. 8-11, 2001 — This paper focuses on a physics-based
explanation of lift. Calculation of lift based on circulation with
artificially imposed Kutta condition is interpreted as a
mathematical model, having limited "real-world" physics, resulting
from the assumption of potential flow. Also the role of viscosity
is discussed. Author's claim is that viscosity is not important for
lift generation.
- Aerodynamics, Aeronautics, and Flight Mechanics,
McCormick, Barnes W., (1979), Chapter 3, John Wiley & Sons,
Inc., New York ISBN 0-471-03032-5.
- Fundamentals of Flight, Richard S. Shevell,
Prentice-Hall International Editions, ISBN 0-13-332917-8 — This
book is primarily intended as a text for a one semester
undergraduate course in mechanical or aeronautical engineering,
although its sections on theory of flight are understandable with a
passing knowledge of calculus and physics.
External links