page_title: Airfoil
section_title: Thin airfoil theory
hierarchical_section_title: Airfoil / Thin airfoil theory
caption_attribution_description: An aerofoil section is well displayed at the wing tip of this Denney Kitfox (G-FOXC), built in 1991. Photographed by Adrian Pingstone in July 2005 at Kemble Airfield, Gloucestershire, England, and released to the public domain.
context_page_description: An airfoil or aerofoil is the cross-sectional shape of a wing, blade, or sail.
  An airfoil-shaped body moving through a fluid produces an aerodynamic force. The component of this force perpendicular to the direction of motion is called lift. The component parallel to the direction of motion is called drag. Subsonic flight airfoils have a characteristic shape with a rounded leading edge, followed by a sharp trailing edge, often with a symmetric curvature of upper and lower surfaces. Foils of similar function designed with water as the working fluid are called hydrofoils.
  The lift on an airfoil is primarily the result of its angle of attack. When oriented at a suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as aerodynamic force and can be resolved into two components: lift and drag. Most foil shapes require a positive angle of attack to generate lift, but cambered airfoils can generate lift at zero angle of attack. This "turning" of the air in the vicinity of the airfoil creates curved streamlines, resulting in lower pressure on one side and higher pressure on the other.
context_section_description: Thin airfoil theory is a simple theory of airfoils that relates angle of attack to lift for incompressible, inviscid flows. It was devised by German-American mathematician Max Munk and further refined by British aerodynamicist Hermann Glauert and others in the 1920s. The theory idealizes the flow around an airfoil as two-dimensional flow around a thin airfoil.  It can be imagined as addressing an airfoil of zero thickness and infinite wingspan.
  Thin airfoil theory was particularly notable in its day because it provided a sound theoretical basis for the following important properties of airfoils in two-dimensional flow:
  on a symmetric airfoil, the center of pressure and aerodynamic center are coincident and lie exactly one quarter of the chord behind the leading edge.
  on a cambered airfoil, the aerodynamic center lies exactly one quarter of the chord behind the leading edge.
  the slope of the lift coefficient versus angle of attack line is  units per radian.
  As a consequence of (3), the section lift coefficient of a symmetric airfoil of infinite wingspan is:
  where  is the section lift coefficient,
   is the angle of attack in radians, measured relative to the chord line.
  (The above expression is also applicable to a cambered airfoil where  is the angle of attack measured relative to the zero-lift line instead of the chord line.)
  Also as a consequence of (3), the section lift coefficient of a cambered airfoil of infinite wingspan is:
  where  is the section lift coefficient when the angle of attack is zero.
  Thin airfoil theory does not account for the stall of the airfoil, which usually occurs at an angle of attack between 10° and 15° for typical airfoils. In the mid-late 2000s, however, a theory predicting the onset of leading-edge stall was proposed by Wallace J. Morris II in his doctoral thesis.  Morris's subsequent refinements contain the details on the current state of theoretical knowledge on the leading-edge stall phenomenon. Morris's theory predicts the critical angle of attack for leading-edge stall onset as the condition at which a global separation zone is predicted in the solution for the inner flow. Morris's theory demonstrates that a subsonic flow about a thin airfoil can be described in terms of an outer region, around most of the airfoil chord, and an inner region, around the nose, that asymptotically match each other. As the flow in the outer region is dominated by classical thin airfoil theory, Morris's equations exhibit many components of thin airfoil theory.
caption: An airfoil section is displayed at the tip of this Denney Kitfox aircraft, built in 1991.
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