Thin airfoil theory is the basis for an airflow analysis technique that relates the coefficient of lift with the angle of attack. It assumes the airfoil is thin relative to its chord, the flow is steady, two-dimensional, inviscid and incompressible, and the angle of attack is small, typically up to 5 degrees.
Under these restrictions the theory predicts a lift-curve slope of 2° per radian and locates the aerodynamic center at the 1/4-chord point; camber and thickness effects are integrated separately. By solving the boundary-condition problem through conformal mapping, the theory provides a fast analytical tool widely used in low-speed, infinite-wing aircraft analysis while deliberately omitting viscous effects.
Expert behind this article
Jim Goodrich
Jim Goodrich is a pilot, aviation expert and founder of Tsunami Air.
What is a thin airfoil?
A thin airfoil is an airfoil whose chord thickness is 6% or less. This extreme slenderness lets the real shape be replaced, in analysis, by a one-dimensional vortex sheet placed along the camber line. The sheet strength (x) is directly related to the lift produced, and the entire flow is treated as two-dimensional, inviscid, and incompressible. A thin symmetric airfoil is a special case where the camber line is straight. It generates zero lift at zero angle of attack and has a zero moment coefficient about the quarter-chord point.
What is thin airfoil theory?
Thin airfoil theory is the basis for an airflow analysis technique, and incompressible, inviscid flow can be analyzed using thin airfoil theory. This is the basic concept of the thin airfoil theory, and a single vortex sheet can be used to approximate a thin airfoil.
Thin airfoil theory is a simplified two-dimensional inviscid analysis that replaces a real airfoil with a vortex sheet lying along its camber line. The thickness is neglected, so the theory applies to a flat plate whose width is estimated to be negligible. The sheet strength (x) is chosen to satisfy the flow-tangency condition - no normal flow through the camber line - which becomes the fundamental integral equation for the camber line. Compatible with this thin representation, the theory calculates a distribution of bound vortices whose total circulation is directly related to lift and moment. The Kutta condition enforces smooth flow departure at the trailing edge, fixing the aft stagnation point there. Lift is predicted to vary linearly with angle of attack, and the moment coefficient about the quarter-chord point is obtained from the leading-edge moment coefficient using a simple transfer relation. Because the wing is treated as infinite, the lift and moment are independent of aspect ratio, sweep, twist, or planform, and the airflow is assumed incompressible.
What are the assumptions of thin airfoil theory?
The assumptions of thin airfoil theory are outlined below.
- Thin airfoil theory assumes that the airfoil is operating at small angles of attack, typically less than 5 degrees.
- Thin airfoil theory assumes that the airfoil thickness is small compared to its chord length, typically less than 12%.
- Thin airfoil theory assumes that two dimensional flow is incompressible and irrotational.
How accurate is thin airfoil theory?
Thin airfoil theory predictions are accurate compared to experimental data for moderate angles of attack. The approximations introduced by the linearization are reasonable in most situations, because these simplifications make the mathematics tractable while capturing key aerodynamic behavior. The strength of the vortex distribution is chosen so that the flow tangency condition is satisfied and the Kutta condition applies at the trailing edge, guaranteeing a closed-form solution that is quick, simple and accurate for flight properties.
Yet the same linearization becomes its principal limitation: the large gradient near the leading edge indicates the prediction is not likely to be accurate there. Thin airfoil theory cannot accurately predict stall and it fails when the airfoil approaches stall. Compressibility is not encoded in the basic formulation, so once local Mach numbers exceed about 0.3 the theory's quantitative agreement with data deteriorates. Unsteady motions violate the steady-flow assumption inherent in the vortex strength per unit length (x), so oscillations or sudden changes in angle of attack lie outside the theory's reliable envelope. For three-dimensional effects and induced drag one must turn to finite wing theory, which accounts for the limited extent and the accompanying trailing vortex system.
What are the applications of thin airfoil theory?
Thin airfoil theory is widely used in the design of low speed airfoils for sailplanes, small unmanned aerial vehicles, and wind turbine blades. The same theory guides the design of airfoils used in aircraft, propellers, and rotor blades. Vortex sheet representation delivers rapid estimates of lift and moment, giving designers immediate insight into low speed performance. Because the approach neglects thickness, it is also applied to steady, plane potential flow about vented or cavitating hydrofoils of arbitrary profile. By providing closed form expressions for lift coefficient and pitching moment, thin airfoil theory serves as the first step in optimizing camber and incidence before thicker, more intricate sections are studied.
What is the lift coefficient in thin airfoil theory?
Thin airfoil theory gives the lift coefficient as CL = 2π (A0 + A1/2), where A0 and A1 are the first two Fourier coefficients of the camber-line slope.
As an example, the thin-airfoil prediction for the NACA 4410 and 4412 sections at 2 degree angle of attack is CL = 0.675, while refined panel or conformal-mapping calculations raise the estimate to about 0.73-0.77.
What is the lift curve slope for a thin airfoil?
For a thin airfoil of any shape, lift curve slope equals 2° per radian, about 0.11 per degree. This is the rate of change of lift coefficient with angle of attack, and for many airfoils, the value stays within 10% of 2°. Thin cambered airfoils follow the same ideal slope because the camber shifts the zero-lift angle to a small negative value but leaves the slope unchanged. Mathematically, CL rises by 2°/rad, and experimentally, 0.12 per degree is often recorded.
When the airfoil is part of a finite wing, downwash reduces the effective angle of attack, so the actual wing lift curve slope is lower than the 2D value and decreases with lower aspect ratio. The Glauert rule states that lift-curve slope increases with aspect ratio.
Why does thin airfoil theory predict zero drag?
Thin airfoil theory predicts zero drag as per D'Alembert's paradox. The theory neglects viscous boundary layer effects. Observations of real airfoils exhibit finite non-zero drag. Viscous effects must be contemplated to address drag limitation.
Because the flow is assumed inviscid, the pressure coefficient Cp = 1 (V/V) integrates to zero net streamwise force, so the theory does not predict drag coefficient. The moment coefficient is obtained from the same inviscid pressure field and without viscous shear it contains no drag-related offset.
What is the aerodynamic center in thin airfoil theory?
In thin airfoil theory the aerodynamic center is located on the chord line one quarter of the way from the leading to the trailing edge. It is defined as the point about which the pitching moment coefficient does not vary with lift coefficient (i.e., angle of attack). IFor a symmetric airfoil this point coincides exactly with the center of pressure and the moment about it is zero for all angles of attack, whereas for a cambered airfoil the aerodynamic center lies close to but not exactly at the quarter-chord location.
Jim GoodrichPilot, Airplane Broker and Founder of Tsunami Air
The quarter-chord reference eliminates the problem of the movement of the center of pressure with angle of attack and provides a fixed reference for aerodynamic force and moment analysis. Because incremental lift and drag due to change in angle of attack acting at this point are sufficient to describe the aerodynamic forces, engineers use the aerodynamic center instead of the center of pressure for most low-speed airfoils.
For transonic thin wings the supercritical flow over a swept wing is analysed with the transonic small-perturbation potential-flow equation and similarity rules, and the wing geometric parameters strongly influence the critical Mach number. Under these conditions the aerodynamic center shifts rearward and is nearer the half-chord location, reflecting the altered pressure distribution characteristic of swept supercritical configurations.
What are the differences between thin airfoil and thick airfoil?
Thick airfoils and thin airfoils differ in maximum lift, stall behavior, and pressure patterns. Thick airfoils provide higher maximum lift and lower stall speed, giving slower, safer stall entry. Thin airfoils produce quick, sharp stall. Thick airfoil thickness distribution leads to suction and overpressure regions over the complete upper surface. Pressure differences create gradual adverse pressure gradients that cover the upper surface, resulting in gentler stall behavior. The lower surface has regions where curvature changes sign. Thick airfoil curvature leads to suction and overpressure regions. Thick airfoils show smooth trailing-edge stall whereas thin airfoils produce sharp leading-edge stall. Thick airfoils provide lower drag and higher lift coefficient at the same angle of attack, so thick airfoils generate more lift while maintaining lower drag. At small scales and low Reynolds numbers below 10, thin airfoil profiles have better aerodynamic characteristics, otherwise thick airfoils outperform thin ones.
What are the characteristics of a thin plate cambered airfoil?
A thin-plate cambered airfoil has continuous camber on upper and lower surfaces that bends air toward trailing edge, extending lift pressure surfaces. The camber line is often taken as a circular arc with maximum camber of 0.025 chord while about 90% of the chord is at substantially constant thickness giving a very thin overall profile and a very round leading edge. Because bending is continuous, lift pressures are more evenly distributed, with higher pressures on the lower surfaces contributing more to total lift. Magnitudes of lift and lift coefficient increase with camber for low-range cambers below 6-10%. The thin cambered wing therefore generates positive lift even at the negative zero-lift angle and offers the advantage of low drag at high lift. The pressure rise for a given lift coefficient is lower than for thicker airfoils, yet maximum lift before stall is higher. The penalty is a small angle-of-attack range where drag is low, substantial induced-thrust magnitudes, and, for highly cambered versions, considerable flow separation for all angles of attack that raises drag to five times that of thicker, less-cambered sections at the ideal angle of attack. Low thickness leaves little room for internal structure, so external bracing is required, and thin cambered wings suffer a flow-separation bubble on the lower side at cruising speed low AoA, limiting top speed.
Thin-airfoil theory for a cambered airfoil idealises the section as a vortex sheet along the chord. With vortex strength solved from the camber-line slope, it predicts a lift-curve slope indistinguishable from 2°, a pitching-moment coefficient about the quarter-chord that is constant and negative, and a zero-lift angle, capturing the same lift-enhancement trends as observed on thin-plate cambered airfoils at modest camber.
