Editing Lift (force) (section)

===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.