Editing Lift (force) (section)

==Mathematical theories of lift==
Mathematical theories of lift are based on continuum fluid mechanics, assuming that air flows as a continuous fluid.<ref>Batchelor (1967), Section 1.2</ref><ref>Thwaites (1958), Section I.2</ref><ref>von Mises (1959), Section I.1</ref> Lift is generated in accordance with the fundamental principles of physics, the most relevant being the following three principles:<ref>"Analysis of fluid flow is typically presented to engineering students in terms of three fundamental principles: conservation of mass, conservation of momentum, and conservation of energy." Charles N. Eastlake ''An Aerodynamicist's View of Lift, Bernoulli, and Newton'' The Physics Teacher Vol. 40, March 2002 {{cite web|url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|title=Archived copy|access-date=10 September 2009|url-status=dead|archive-url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|archive-date=April 11, 2009}}</ref>
* [[Conservation of momentum]], which is a consequence of [[Newton's laws of motion]], especially Newton's second law which relates the net [[force]] on an element of air to its rate of [[momentum]] change,
* [[Continuity equation#Fluid dynamics|Conservation of mass]], including the assumption that the airfoil's surface is impermeable for the air flowing around, and
* [[Conservation of energy]], which says that energy is neither created nor destroyed.

Because an airfoil affects the flow in a wide area around it, the conservation laws of mechanics are embodied in the form of [[partial differential equation]]s combined with a set of [[boundary condition]] requirements which the flow has to satisfy at the airfoil surface and far away from the airfoil.<ref>White (1991), Chapter 1</ref>

To predict lift requires solving the equations for a particular airfoil shape and flow condition, which generally requires calculations that are so voluminous that they are practical only on a computer, through the methods of [[computational fluid dynamics]] (CFD). Determining the net aerodynamic force from a CFD solution requires "adding up" ([[integration (mathematics)|integrating]]) the forces due to pressure and shear determined by the CFD over every surface element of the airfoil as described under "[[#Pressure_integration|pressure integration]]".

The [[Navier–Stokes equations]] (NS) provide the potentially most accurate theory of lift, but in practice, capturing the effects of turbulence in the boundary layer on the airfoil surface requires sacrificing some accuracy, and requires use of the [[Reynolds-averaged Navier–Stokes equations]] (RANS). Simpler but less accurate theories have also been developed.

===Navier–Stokes (NS) equations===

These equations represent conservation of mass, Newton's second law (conservation of momentum), conservation of energy, the [[Viscosity#Newtonian and non-Newtonian fluids|Newtonian law for the action of viscosity]], the [[Fourier heat conduction equation#Fourier.27s law|Fourier heat conduction law]], an [[equation of state]] relating density, temperature, and pressure, and formulas for the viscosity and thermal conductivity of the fluid.<ref>Batchelor (1967), Chapter 3</ref><ref>Aris (1989)</ref>

In principle, the NS equations, combined with boundary conditions of no through-flow and no slip at the airfoil surface, could be used to predict lift with high accuracy in any situation in ordinary atmospheric flight. However, airflows in practical situations always involve turbulence in the boundary layer next to the airfoil surface, at least over the aft portion of the airfoil. Predicting lift by solving the NS equations in their raw form would require the calculations to resolve the details of the turbulence, down to the smallest eddy. This is not yet possible, even on the most powerful computer.<ref name=Spalart>Spalart, Philippe R. (2000) Amsterdam, the Netherlands. Elsevier Science Publishers.</ref> So in principle the NS equations provide a complete and very accurate theory of lift, but practical prediction of lift requires that the effects of turbulence be modeled in the RANS equations rather than computed directly.

===Reynolds-averaged Navier–Stokes (RANS) equations===

These are the NS equations with the turbulence motions averaged over time, and the effects of the turbulence on the time-averaged flow represented by [[turbulence modeling]] (an additional set of equations based on a combination of [[dimensional analysis]] and empirical information on how turbulence affects a boundary layer in a time-averaged average sense).<ref>White (1991), Section 6-2</ref><ref>Schlichting(1979), Chapter XVIII</ref> A RANS solution consists of the time-averaged velocity vector, pressure, density, and temperature defined at a dense grid of points surrounding the airfoil.

The amount of computation required is a minuscule fraction (billionths)<ref name=Spalart /> of what would be required to resolve all of the turbulence motions in a raw NS calculation, and with large computers available it is now practical to carry out RANS calculations for complete airplanes in three dimensions. Because turbulence models are not perfect, the accuracy of RANS calculations is imperfect, but it is adequate for practical aircraft design. Lift predicted by RANS is usually within a few percent of the actual lift.

===Inviscid-flow equations (Euler or potential)===

The [[Euler equations (fluid dynamics)|Euler equations]] are the NS equations without the viscosity, heat conduction, and turbulence effects.<ref>Anderson (1995)</ref> As with a RANS solution, an Euler solution consists of the velocity vector, pressure, density, and temperature defined at a dense grid of points surrounding the airfoil. While the Euler equations are simpler than the NS equations, they do not lend themselves to exact analytic solutions.

Further simplification is available through [[potential flow]] theory, which reduces the number of unknowns to be determined, and makes analytic solutions possible in some cases, as described below.

Either Euler or potential-flow calculations predict the pressure distribution on the airfoil surfaces roughly correctly for angles of attack below stall, where they might miss the total lift by as much as 10–20%. At angles of attack above stall, inviscid calculations do not predict that stall has happened, and as a result they grossly overestimate the lift.

In potential-flow theory, the flow is assumed to be [[irrotational]], i.e. that small fluid parcels have no net rate of rotation. Mathematically, this is expressed by the statement that the [[Curl (mathematics)|curl]] of the velocity vector field is everywhere equal to zero. Irrotational flows have the convenient property that the velocity can be expressed as the gradient of a scalar function called a [[potential]]. A flow represented in this way is called potential flow.<ref>"...whenever the velocity field is irrotational, it can be expressed as the gradient of a scalar function we
call a velocity potential φ: V = ∇φ. The existence of a velocity potential can greatly simplify the analysis of inviscid flows by way of potential-flow theory..." Doug McLean ''Understanding Aerodynamics: Arguing from the Real Physics'' p. 26 Wiley {{cite book|title=Understanding Aerodynamics|doi=10.1002/9781118454190.ch3|year=2012|page=13|chapter = Continuum Fluid Mechanics and the Navier–Stokes Equations|isbn = 9781118454190}}</ref><ref>Elements of Potential Flow California State University Los Angeles {{cite web|url=http://instructional1.calstatela.edu/cwu/me408/Slides/PotentialFlow/PotentialFlow.htm|title=Faculty Web Directory|access-date=26 July 2012|url-status=dead|archive-url=https://web.archive.org/web/20121111220110/http://instructional1.calstatela.edu/cwu/me408/Slides/PotentialFlow/PotentialFlow.htm|archive-date=November 11, 2012}}</ref><ref>Batchelor(1967), Section 2.7</ref><ref>Milne-Thomson(1966), Section 3.31</ref>

In potential-flow theory, the flow is assumed to be incompressible. Incompressible potential-flow theory has the advantage that the equation ([[Laplace's equation]]) to be solved for the potential is [[Linear#Physics|linear]], which allows solutions to be constructed by [[Superposition principle|superposition]] of other known solutions. The incompressible-potential-flow equation can also be solved by [[conformal mapping]], a method based on the theory of functions of a complex variable. In the early 20th century, before computers were available, conformal mapping was used to generate solutions to the incompressible potential-flow equation for a class of idealized airfoil shapes, providing some of the first practical theoretical predictions of the pressure distribution on a lifting airfoil.

A solution of the potential equation directly determines only the velocity field. The pressure field is deduced from the velocity field through Bernoulli's equation.

[[Image:Airfoil Kutta condition.jpg|thumb|right|400px|Comparison of a non-lifting flow pattern around an airfoil; and a lifting flow pattern consistent with the Kutta condition in which the flow leaves the trailing edge smoothly]]

Applying potential-flow theory to a lifting flow requires special treatment and an additional assumption. The problem arises because lift on an airfoil in inviscid flow requires [[Circulation (fluid dynamics)|circulation]] in the flow around the airfoil (See "[[#Circulation_and_the_Kutta–Joukowski_theorem|Circulation and the Kutta–Joukowski theorem]]" below), but a single potential function that is continuous throughout the domain around the airfoil cannot represent a flow with nonzero circulation. The solution to this problem is to introduce a [[branch cut]], a curve or line from some point on the airfoil surface out to infinite distance, and to allow a jump in the value of the potential across the cut. The jump in the potential imposes circulation in the flow equal to the potential jump and thus allows nonzero circulation to be represented. However, the potential jump is a free parameter that is not determined by the potential equation or the other boundary conditions, and the solution is thus indeterminate. A potential-flow solution exists for any value of the circulation and any value of the lift. One way to resolve this indeterminacy is to impose the [[Kutta condition]],<ref>Clancy (1975), Section 4.8</ref><ref>Anderson(1991), Section 4.5</ref> which is that, of all the possible solutions, the physically reasonable solution is the one in which the flow leaves the trailing edge smoothly. The streamline sketches illustrate one flow pattern with zero lift, in which the flow goes around the trailing edge and leaves the upper surface ahead of the trailing edge, and another flow pattern with positive lift, in which the flow leaves smoothly at the trailing edge in accordance with the Kutta condition.

===Linearized potential flow===

This is potential-flow theory with the further assumptions that the airfoil is very thin and the angle of attack is small.<ref>Clancy(1975), Sections 8.1–8</ref> The linearized theory predicts the general character of the airfoil pressure distribution and how it is influenced by airfoil shape and angle of attack, but is not accurate enough for design work. For a 2D airfoil, such calculations can be done in a fraction of a second in a spreadsheet on a PC.

===Circulation and the Kutta–Joukowski theorem===
[[Image:Circulation-around-aerofoil.svg|thumb|right|300px|Circulation component of the flow around an airfoil]]

When an airfoil generates lift, several components of the overall velocity field contribute to a net circulation of air around it: the upward flow ahead of the airfoil, the accelerated flow above, the decelerated flow below, and the downward flow behind.

The circulation can be understood as the total amount of "spinning" (or [[vorticity]]) of an inviscid fluid around the airfoil.

The [[Kutta–Joukowski theorem]] relates the lift per unit width of span of a two-dimensional airfoil to this circulation component of the flow.<ref name="Clancy 1975, Section 4.5"/><ref>von Mises (1959), Section VIII.2</ref><ref>Anderson(1991), Section 3.15</ref> It is a key element in an explanation of lift that follows the development of the flow around an airfoil as the airfoil starts its motion from rest and a [[starting vortex]] is formed and left behind, leading to the formation of circulation around the airfoil.<ref>Prandtl and Tietjens (1934)</ref><ref>Batchelor (1967), Section 6.7</ref><ref>Gentry (2006)</ref> Lift is then inferred from the Kutta-Joukowski theorem. This explanation is largely mathematical, and its general progression is based on logical inference, not physical cause-and-effect.<ref>McLean (2012), Section 7.2.1</ref>

The Kutta–Joukowski model does not predict how much circulation or lift a two-dimensional airfoil produces. Calculating the lift per unit span using Kutta–Joukowski requires a known value for the circulation. In particular, if the Kutta condition is met, in which the rear stagnation point moves to the airfoil trailing edge and attaches there for the duration of flight, the lift can be calculated theoretically through the conformal mapping method.

The lift generated by a conventional airfoil is dictated by both its design and the flight conditions, such as forward velocity, angle of attack and air density. Lift can be increased by artificially increasing the circulation, for example by boundary-layer blowing or the use of [[blown flap]]s. In the [[Flettner rotor]] the entire airfoil is circular and spins about a spanwise axis to create the circulation.