Lift (force)

# Lift (force)

Overview
A fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

flowing past the surface of a body exerts a surface force
Surface force
Surface force denoted fs is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed in to two perpendicular components: pressure and stress forces....

on it. Lift is the component of this force that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the oncoming flow direction. It contrasts with the drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

force, which is the component of the surface force parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

to the flow direction. If the fluid is air, the force is called an aerodynamic force.
Discussion

Recent Discussions
Encyclopedia
A fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

flowing past the surface of a body exerts a surface force
Surface force
Surface force denoted fs is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed in to two perpendicular components: pressure and stress forces....

on it. Lift is the component of this force that is perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the oncoming flow direction. It contrasts with the drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

force, which is the component of the surface force parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

to the flow direction. If the fluid is air, the force is called an aerodynamic force.

## Overview

Lift is commonly associated with the wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...

of a fixed-wing aircraft
Fixed-wing aircraft
A fixed-wing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixed-wing aircraft are distinct from rotary-wing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...

, although lift is also generated by propeller
Propeller (aircraft)
Aircraft propellers or airscrews convert rotary motion from piston engines or turboprops to provide propulsive force. They may be fixed or variable pitch. Early aircraft propellers were carved by hand from solid or laminated wood with later propellers being constructed from metal...

s; kites
Kite types
Kites are tethered flying objects which fly by using aerodynamic lift, requiring wind, , for generation of airflow over the lifting surfaces.-Kite types:...

; helicopter rotor
Helicopter rotor
A helicopter main rotor or rotor system is a type of fan that is used to generate both the aerodynamic lift force that supports the weight of the helicopter, and thrust which counteracts aerodynamic drag in forward flight...

s; rudder
Rudder
A rudder is a device used to steer a ship, boat, submarine, hovercraft, aircraft or other conveyance that moves through a medium . On an aircraft the rudder is used primarily to counter adverse yaw and p-factor and is not the primary control used to turn the airplane...

s, sail
Sail
A sail is any type of surface intended to move a vessel, vehicle or rotor by being placed in a wind—in essence a propulsion wing. Sails are used in sailing.-History of sails:...

s and keel
Keel
In boats and ships, keel can refer to either of two parts: a structural element, or a hydrodynamic element. These parts overlap. As the laying down of the keel is the initial step in construction of a ship, in British and American shipbuilding traditions the construction is dated from this event...

s on sailboat
Sailboat
A sailboat or sailing boat is a boat propelled partly or entirely by sails. The term covers a variety of boats, larger than small vessels such as sailboards and smaller than sailing ships, but distinctions in the size are not strictly defined and what constitutes a sailing ship, sailboat, or a...

s; hydrofoil
Hydrofoil
A hydrofoil is a foil which operates in water. They are similar in appearance and purpose to airfoils.Hydrofoils can be artificial, such as the rudder or keel on a boat, the diving planes on a submarine, a surfboard fin, or occur naturally, as with fish fins, the flippers of aquatic mammals, the...

s; wings
Wing (automotive)
(for the panels around the wheels of a car see Fender (vehicle)A wing in this context is an aerodynamic device intended to generate downforce on an automobile. The first production car to feature a rear wing was the 1969 mercury cyclone spoiler, soon followed by the drastically larger Superbird and...

on auto racing
Auto racing
Auto racing is a motorsport involving the racing of cars for competition. It is one of the world's most watched televised sports.-The beginning of racing:...

cars; wind turbine
Wind turbine
A wind turbine is a device that converts kinetic energy from the wind into mechanical energy. If the mechanical energy is used to produce electricity, the device may be called a wind generator or wind charger. If the mechanical energy is used to drive machinery, such as for grinding grain or...

s and other streamlined objects. While the common meaning of the word "lift" assumes that lift opposes gravity, lift in its technical sense can be in any direction since it is defined with respect to the direction of flow rather than to the direction of gravity. When an aircraft is flying straight and level (cruise
Cruise (flight)
Cruise is the level portion of aircraft travel where flight is most fuel efficient. It occurs between ascent and descent phases and is usually the majority of a journey. Technically, cruising consists of heading changes only at a constant airspeed and altitude...

) most of the lift opposes gravity. However, when an aircraft is climb
Climb
thumb|right|An [[Embraer ERJ 145]] climbingIn aviation, the term climb refers both to the actual operation of increasing the altitude of an aircraft and to the logical phase of a typical flight following take-off and preceding the cruise, during which an increase in altitude to a predetermined...

ing, descending
Descent (aircraft)
A descent during air travel is any portion where an aircraft decreases altitude, and is the opposite of an ascent or climb. Descents are an essential component of an approach to landing...

, 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
Aerobatics
Aerobatics is the practice of flying maneuvers involving aircraft attitudes that are not used in normal flight. Aerobatics are performed in airplanes and gliders for training, recreation, entertainment and sport...

, or on the wing on a racing car. In this last case, the term downforce
Downforce
Downforce is a downwards thrust created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip....

is often used. Lift may also be horizontal, for instance on a sail
Sail
A sail is any type of surface intended to move a vessel, vehicle or rotor by being placed in a wind—in essence a propulsion wing. Sails are used in sailing.-History of sails:...

on a sailboat
Sailboat
A sailboat or sailing boat is a boat propelled partly or entirely by sails. The term covers a variety of boats, larger than small vessels such as sailboards and smaller than sailing ships, but distinctions in the size are not strictly defined and what constitutes a sailing ship, sailboat, or a...

.

An airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

is a streamlined shape that is capable of generating significantly more lift than drag. Non-streamlined objects such as bluff bodies and plates (not parallel to the flow) may also generate lift when moving relative to the fluid.

## 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. For example, there are explanations based directly on Newton’s laws of motion and explanations based on Bernoulli’s principle. Both principles can be used to explain lift, but each appeals to a different audience. This article will start with the simplest explanation; more complicated and alternative explanations will follow.

#### Deflection

One way to understand the generation of lift is to observe that the air is deflected as it passes the airfoil. Since the foil must exert a force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

on the air to change its direction, the air must exert a force of equal magnitude but opposite direction on the foil. In the case of an airplane wing, the wing exerts a downward force on the air and the air exerts an upward force on the wing.

This explanation relies on the second and third of Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

: The net force on an object is equal to its rate of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

change
, and: To every action there is an equal and opposite reaction.

Another way to describe deflection is to say that the air "turns" as it passes the airfoil and follows a path that is curved. When airflow changes direction, a force is generated.

#### Pressure differences

Lift may also be described in terms of air pressure: pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

is the normal force per unit area. Wherever there is net force there is also a pressure difference, thus deflection/flow turning indicates the presence of a net force and a pressure difference. This pressure difference implies the average pressure on the upper surface of the wing is lower than the average pressure on the underside.

#### Flow on both sides of the wing

In the picture above, observe that the air is turned both above and below the wing so both the upper and lower surface contribute to the flow turning and therefore the lift. In fact, for typical airfoils at subsonic speeds the top surface contributes more flow turning than the bottom surface, and the pressure deviation along the top is significantly larger than along the bottom. A common explanation describes lift as merely the result of the air molecules bouncing off the lower surface of the wing, but since this ignores the airflow around the top of the wing it usually leads to incorrect results. However, at hypersonic speeds, this model becomes applicable.

#### Criticisms of deflection/turning

• While the theory correctly reasons that deflection implies that there must be a force on the wing, it does not explain why the air is deflected. Intuitively, one can say that the air follows the curve of the foil, but this is not very rigorous or precise.
• The theory, while correct in as far as it goes, is not sufficient to allow one to do engineering. Fluid stresses – including pressure – need to be related to the fluid motion (e.g. through constitutive equation
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...

s). Thus, textbooks on aerodynamics use more complex models to provide a full description of lift.

### A more rigorous physical description

Explaining lift while considering all of the principles involved is a complex task and is not easily simplified. Lift is generated in accordance with the fundamental principles of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. The most relevant physics reduce to three principles:
• Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

, especially Newton's second law which relates the net force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

on an element of air to its rate of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

change,
• conservation of mass, including the common assumption that the airfoil's surface is impermeable for the air flowing around, and
• an expression relating the fluid stresses
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

(consisting of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

and shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

components) to the properties of the flow.

In the last principle, the pressure depends on the other flow properties, such as its mass density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, through the (thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

) equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

, while the shear stresses are related to the flow through the air's viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called 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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

. The particular form of Bernoulli's equation found depends on the equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

used. At low Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

s, compressibility effects may be neglected, resulting in an incompressible flow
Incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...

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
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

evolution by the Euler equations—along a streamline.

In order to explain lift as it applies to an airplane wing, consider the incompressible flow around a 2-D, symmetric airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

at positive angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...

in a uniform freestream. 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

If one takes the experimentally observed flow around an airfoil as a starting point, then lift can be explained in terms of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

s using Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

(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
NACA airfoil
The NACA airfoils are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics . The shape of the NACA airfoils is described using a series of digits following the word "NACA." The parameters in the numerical code can be entered into equations to precisely...

airfoil computed using potential flow theory
Potential flow
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...

, 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
Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

. 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 parcel
Air parcel
In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, while—in a compressible flow—its volume may...

s 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 freestream is considered to be lift; the component parallel to the freestream is drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

. 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
Downwash
In aeronautics downwash is the air forced down by the aerodynamic action of a wing or helicopter rotor blade in motion, as part of the process of producing lift....

. Measuring the momentum transferred to the downwash is another way to determine the amount of lift on the airfoil.

#### Flowfield formation

The last section shows that one can use Bernoulli's principle to explain lift, assuming one knows the airflow in the vicinity of the airfoil. In attempting to explain why the air flows the way it does (e.g. why the flow follows the upper surface of the airfoil and why the streamtubes change size), 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
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called 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 point
Stagnation point
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero. Stagnation points exist at the surface of objects in the flow field, where the fluid is brought to rest by the object...

s 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
Static pressure
In fluid mechanics the term static pressure has several uses:* In the design and operation of aircraft, static pressure is the air pressure in the aircraft’s static pressure system....

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
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

, 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
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

, 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
Starting vortex
The starting vortex is a vortex which forms in the air adjacent to the trailing edge of an airfoil as it is accelerated from rest in a fluid. It leaves the airfoil , and remains stationary in the flow...

. 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
Starting vortex
The starting vortex is a vortex which forms in the air adjacent to the trailing edge of an airfoil as it is accelerated from rest in a fluid. It leaves the airfoil , and remains stationary in the flow...

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.

#### "Popular" explanation based on equal transit-time

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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

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.

#### 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
Entrainment (hydrodynamics)
Entrainment is the movement of one fluid by another.One fluid moving in another can push or pull the other along with it. Eductors or eductor-jet pumps are an excellent example. They are used onboard many ships to pump flooded out compartments in the event of an accident. Seawater is pumped to...

of ambient air into the flow. The effect is named for Henri Coandă
Henri Coanda
Henri Marie Coandă was a Romanian inventor, aerodynamics pioneer and builder of an experimental aircraft, the Coandă-1910 described by Coandă in the mid-1950s as the world's first jet, a controversial claim disputed by some and supported by others...

, the Romania
Romania
Romania is a country located at the crossroads of Central and Southeastern Europe, on the Lower Danube, within and outside the Carpathian arch, bordering on the Black Sea...

n 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"
Antonov An-72
The Antonov An-72 is a transport aircraft developed by Antonov in the former Soviet Union. It was designed as a STOL transport and intended as a replacement for the Antonov An-26, but variants have found success as commercial freighters.The An-72 gets its nickname, Cheburashka, from the large...

, use the exhaust from top-mounted jet engines flowing over the wing to enhance lift, as did the Boeing YC-14
Boeing YC-14
The Boeing YC-14 was a twin-engine short take-off and landing tactical transport. It was Boeing's entrant into the United States Air Force's Advanced Medium STOL Transport competition, which aimed to replace the Lockheed C-130 Hercules as the USAF's standard STOL tactical transport...

and the McDonnell Douglas YC-15
McDonnell Douglas YC-15
|-See also:-References:NotesBibliography* Green, William. The Observer's Book of Aircraft. London. Frederick Warne & Co. Ltd., 1976. ISBN 0-7232-1553-7....

.
The effect is also used in high-lift devices such as a blown flap
Blown flap
Blown flaps are a powered aerodynamic high-lift device invented by the British and used on the wings of certain aircraft to improve low-speed lift during takeoff and landing. The process is sometimes called a boundary layer control system . They were a popular design feature in the 1960s, but fell...

.

More broadly, some consider the effect to include the tendency of any fluid boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

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
Jef Raskin was an American human-computer interface expert best known for starting the Macintosh project for Apple in the late 1970s.-Early years and education:...

, 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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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
Camber (aerodynamics)
Camber, in aeronautics and aeronautical engineering, is the asymmetry between the top and the bottom surfaces of an aerofoil. An aerofoil that is not cambered is called a symmetric aerofoil...

at zero angle of attack, the upper surface has a greater area (that is, the interior area of the airfoil above the chordline
Chord (aircraft)
In aeronautics, chord refers to the imaginary straight line joining the trailing edge and the center of curvature of the leading edge of the cross-section of an airfoil...

) 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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

, 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.

### 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:

where
• L is lift force,
• ρ is air density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

• v is true airspeed
True airspeed
True airspeed of an aircraft is the speed of the aircraft relative to the airmass in which it is flying. True airspeed is important information for accurate navigation of an aircraft.-Performance:...

,
• A is planform
Planform
In aviation, a planform is the shape and layout of a fixed-wing aircraft's fuselage and wing. Of all the myriad planforms used, they can typically be grouped into those used for low-speed flight, found on general aviation aircraft, and those used for high-speed flight, found on many military...

area, and
• is the lift coefficient at the desired angle of attack, Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

, and Reynolds number

### Kutta–Joukowski theorem

Lift can be calculated using potential flow
Potential flow
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...

theory by imposing a circulation. It is often used by practising aerodynamicists as a convenient quantity in calculations, for example thin-airfoil theory and lifting-line theory
Lifting-line theory
Lifting-line theory or Lanchester-Prandtl wing theory was published by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk on the problem of a useful mathematical tool for examining lift from "real world" wings....

.

The circulation is the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

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 can be calculated using the Kutta–Joukowski theorem
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

:

where is the air density, is the free-stream airspeed. Kelvin's circulation theorem
Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem states In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time. The theorem was developed by William Thomson, 1st Baron Kelvin...

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
Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

, 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
Magnus effect
The Magnus effect is the phenomenon whereby a spinning object flying in a fluid creates a whirlpool of fluid around itself, and experiences a force perpendicular to the line of motion...

, which is a lift force generated by a spinning cylinder in a freestream. 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
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

differences above and below the wing, which can be related to velocity changes by Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

.

The total lift force is the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

of vertical pressure forces over the entire wetted surface area of the wing:

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 Dp (which includes induced drag
Lift-induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars...

in a 3D wing). If we use the spanwise vector j, we obtain the side force Y.

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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

, which can be important in the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

and to predict friction drag
Parasitic drag
Parasitic drag is drag caused by moving a solid object through a fluid medium . Parasitic drag is made up of many components, the most prominent being form drag...

, which is the other component of the total drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

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
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

and scientist Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

, though it was previously understood by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

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
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's 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
Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

must be applied to simulate the effects of 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
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

and conservation of momentum.

## Lift forces on bluff bodies

The flow around bluff bodies may also generate lift, besides a strong drag force. This lift may be steady, or it may oscillate
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

due to vortex shedding
Vortex shedding
Vortex shedding is an unsteady flow that takes place in special flow velocities . In this flow, vortices are created at the back of the body and detach periodically from either side of the body. See Von Kármán vortex street.Vortex shedding is caused when a fluid flows past a blunt object...

. Interaction of the object's flexibility with the vortex shedding may enhance the effects of fluctuating lift and cause vortex-induced vibrations. For instance, the flow around a circular cylinder generates a Kármán vortex street: vortices
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

is characterised by the dimensionless Strouhal number
Strouhal number
In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind...

, 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
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

or strong spanwise correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

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 chimney
Chimney
A chimney is a structure for venting hot flue gases or smoke from a boiler, stove, furnace or fireplace to the outside atmosphere. Chimneys are typically vertical, or as near as possible to vertical, to ensure that the gases flow smoothly, drawing air into the combustion in what is known as the...

s, if not properly taken care of in the design.

• Aerodynamic force
• Banked turn
Banked turn
A banked turn is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to its path....

• Circulation control wing
Circulation control wing
A circulation control wing is a form of high-lift device for use on the main wing of an aircraft to increase the lift coefficient. CCW technology has been in the research and development phase for over sixty years, and the early models were called blown flaps.The CCW works by increasing the...

• Downforce
Downforce
Downforce is a downwards thrust created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip....

• Drag (physics)
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

• Foil (fluid mechanics)
Foil (fluid mechanics)
A foil is a solid object with a shape such that when placed in a moving fluid at a suitable angle of attack the lift is substantially larger than the drag...

• Küssner effect
• Kutta condition
Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

• Kutta–Joukowski theorem
Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

• Lift-induced drag
Lift-induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars...

• Lift-to-drag ratio
Lift-to-drag ratio
In aerodynamics, the lift-to-drag ratio, or L/D ratio, is the amount of lift generated by a wing or vehicle, divided by the drag it creates by moving through the air...

• Lifting-line theory
Lifting-line theory
Lifting-line theory or Lanchester-Prandtl wing theory was published by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk on the problem of a useful mathematical tool for examining lift from "real world" wings....

• 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.

• 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.

• "Observation of Perfect Potential Flow in Superfluid", Paul P. Craig and John R. Pellam (1957) Physical Review 108(5), pp. 1109–1112, – Experiments under superfluidity conditions, resulting in the vanishing of lift in inviscid flow since the Kutta condition
Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

no longer is satisfied.

• "Flight without Bernoulli" Chris Waltham Vol. 36, Nov. 1998 THE PHYSICS TEACHER – using a physical model relying only on Newton’s second law, the author presents a rigorous fluid dynamical treatment of flight. http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf

• "Bernoulli and Newton in Fluid Mechanics" Norman F. Smith The Physics Teacher November 1972 Volume 10, Issue 8, pp. 451 – Original paper criticizing the "popular" explanation of lift and explaining it in terms of Newton's laws http://tpt.aapt.org/resource/1/phteah/v10/i8