Definitions

# Lift (force)

In the context of a fluid flow relative to a body, the lift force is the component of the aerodynamic force that is perpendicular to the flow direction. It contrasts with the drag force, which is the parallel component of the aerodynamic force.

Lift is commonly associated with the wing of an aircraft, although lift is also generated by rotors on helicopters, sails and keels on sailboats, hydrofoils, wing on auto racing cars, and wind turbines. While common meanings of the word "lift#English" suggest that lift opposes gravity, aerodynamic lift can be in any direction. When an aircraft is in cruise for example, lift does oppose gravity. However, when the 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.

The mathematical equations describing the generation of lift forces have been well established since the Wright Brothers experimentally determined a reasonably precise value for Smeaton's Smeaton coefficient more than 100 years ago, but the practical explanation of what those equations mean is still controversial, with persistent misinformation and pervasive misunderstanding.

## Physical description of lift on an airfoil

Lift is generated in accordance with the fundamental principles of physics such as Newton's laws of motion, Bernoulli's principle, conservation of mass and the balance of momentum (where the latter is the fluid dynamics version of Newton's second law). Each of these principles can be used to explain lift on an airfoil. As a result, there are numerous different explanations with different levels of rigour and complexity. For example, there is an explanation based 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.

To attempt a physical explanation of lift as it applies to an airplane, consider the 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

If one assumes that the flow naturally follows the shape of an airfoil, as is the usual observation, then the explanation of lift is rather simple and can be explained primarily in terms of pressures using Bernoulli's principle (which can be derived from Newton's second law) and conservation of mass, following the development by John D. Anderson in Introduction to Flight.

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 streamlines. By definition, fluid never crosses a streamline; 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 that intersects the airfoil on the lower surface, typically near to the leading edge.

The upper stream tube constricts as it flows up and around the airfoil, the so-called upwash. From the conservation of mass, the flow speed must increase as the area of the stream tube decreases. Relatively speaking, the bottom of the airfoil presents less of an obstruction to the free stream, and often expands as the flow travels around the airfoil, slowing the flow below the airfoil. (Contrary to the equal transit-time explanation of lift, there is no requirement that particles that split as they travel over the airfoil meet at the trailing edge. It is typically the case that the particle traveling over the upper surface will reach the trailing edge long before the one 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.

### Stages of lift production

In attempting to explain why the flow follows the upper surface of the airfoil, the situation gets considerably more complex. To offer a more complete physical picture of lift, 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.

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 (a feature of the flow known as 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.

## Methods of determining lift

### 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:

$L = oint pmathbf\left\{n\right\} cdotmathbf\left\{k\right\} ; mathrm\left\{d\right\}A,$

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 Dp (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 pmathbf{n} cdotmathbf{i} ; mathrm{d}A,
` [1.2ex]`
Y &= oint pmathbf{n} cdotmathbf{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 Dp.

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. Erroneous assumptions concerning velocity, e.g., that two parcels of air separated at the front of the wing must meet up again at the back of the wing, are commonly found.

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.

## Mathematical approximations

### 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 $L\text{'}$ can be calculated using the Kutta–Joukowski theorem:

$L\text{'} = -rho V Gamma$

where $rho$ is the air density, $V$ is the free-stream airspeed. The Helmholtz theorem states that circulation is conserved; put simply this 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 separate at different points between top and bottom. The asymmetric separation then produces a circulation in the outer inviscid flow.

### 1900 lift equation

The lift equation used by the Wright brothers was due to John Smeaton. It has the form:

$L = k V^2 A C_l ,$

where:

$L$ is the lift
$k$ is the Smeaton coefficient- 0.005 (the drag of a 1 square foot plate at 1 mph)
$C_l$ is the lift coefficient (the lift relative to the drag of a plate of the same area)
$A$ is the area in square feet

The Wright brothers determined with wind tunnels that the Smeaton coefficient was incorrect and should have been 0.0033.

## Alternative Explanations

### Equal transit-time

One misconception encountered in a number of popular explanations of lift is the "equal transit time" fallacy. This fallacy assumes that the parcels of air that are divided above and below an airfoil must rejoin behind it. The fallacy 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. 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.

A further flaw in this explanation is that it requires an airfoil to have thickness and curvature in order to create lift. In fact, thin flat plate wings and sails create lift under a range of angles of attack. If lift were solely a result of shape, then it would not be possible to fly inverted.

This explanation has gained currency by repetition in populist (rather than technical) books. At least one common pilot training book depicts the equal transit fallacy, adding to the confusion.

### 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 first known uses is in his patent for a high-lift device that uses a fan of gas exiting at high pressure from an internal compressor. This circular spray is 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 is caused by the difference between this pressure and that on the bottom of the craft. Two Russian 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 prototype 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 that involving a 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 Coanda 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 the lower surface the flow is quiescent. The physics of this demonstration are very different from that off the general flow over the wing. The usage in this sense is largely seen in popular references on aerodynamics. Those in the aerodynamics field generally consider the Coanda effect in the more limited sense above and use viscosity to explain why the boundary layer attaches to the surface of a wing.