Math 2.0

Fluid mechanics Definition

Thin airfoil theory

Builds on Potential flowCirculationKutta conditionKutta–Joukowski theoremAirfoil equationCauchy principal value

Definition

Consider a steady, two-dimensional, incompressible, inviscid and irrotational flow of speed VV past an airfoil of chord cc placed along 0≤x≤c0 \le x \le c. Let z=zc(x)z = z_c(x) be its camber line and α\alpha its angle of attack, and assume that α\alpha, zc/cz_c / c, zc′z_c' and the relative thickness are small. Thin airfoil theory is the linearised model of this flow in which:

  1. the airfoil is replaced by a vortex sheet on the segment 0≤x≤c0 \le x \le c, with strength γ(x)\gamma(x) per unit length;
  2. the flow is tangent to the camber line, which to first order reads 12π PV ⁣∫0cγ(ξ)x−ξ dξ=V(α−zc′(x)),0<x<c,\frac{1}{2\pi}\, \mathrm{PV}\!\int_0^c \frac{\gamma(\xi)}{x - \xi}\, d\xi = V\bigl(\alpha - z_c'(x)\bigr), \quad 0 < x < c, where PV\mathrm{PV} is the Cauchy principal value;
  3. the Kutta condition γ(c)=0\gamma(c) = 0 holds at the trailing edge.

The lift per unit span is L′=ρVΓL' = \rho V \Gamma, with Γ=∫0cγ(ξ) dξ\Gamma = \int_0^c \gamma(\xi)\, d\xi the circulation and ρ\rho the density, as in the Kutta–Joukowski theorem.

Remarks

Example: for a flat plate, zc=0z_c = 0. With x=c2(1−cos⁡θ)x = \frac{c}{2}(1 - \cos\theta), the function γ(θ)=2αV1+cos⁡θsin⁡θ\gamma(\theta) = 2\alpha V \frac{1 + \cos\theta}{\sin\theta} solves condition 2 and vanishes at θ=π\theta = \pi, which is x=cx = c. Then Γ=παcV\Gamma = \pi \alpha c V, and the lift coefficient is cl=L′/(12ρV2c)=2παc_l = L' / (\frac{1}{2}\rho V^2 c) = 2\pi\alpha. For a cambered section the result is cl=2π(α−α0)c_l = 2\pi(\alpha - \alpha_0), where the zero-lift angle α0\alpha_0 depends only on the camber line. Non-example: a thick section at a large angle of attack does not fit the model, because the small-perturbation assumptions fail, and real flow separates there. After scaling the chord to [−1,1][-1, 1], condition 2 is the airfoil equation.

Reading path

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Cauchy principal value, concept, no evidence levelCauchy principal valueconceptCirculation, concept, no evidence levelCirculationconceptComplex potential, concept, no evidence levelComplex potentialconceptFinite Hilbert transform, concept, no evidence levelFinite Hilbert transf…conceptKutta condition, concept, no evidence levelKutta conditionconceptPotential flow, concept, no evidence levelPotential flowconceptSingular integral equation, concept, no evidence levelSingular integral equ…conceptKutta–Joukowski theorem, statement, no evidence levelKutta–Joukowski theor…statementAirfoil equation, example, no evidence levelAirfoil equationexampleThin airfoil theory, definition, no evidence levelThin airfoil theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cauchy principal value concept
  2. Circulation concept
  3. Complex potential concept
  4. Finite Hilbert transform concept
  5. Kutta condition concept
  6. Potential flow concept
  7. Singular integral equation concept
  8. Kutta–Joukowski theorem statement
  9. Airfoil equation example
  10. Thin airfoil theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Airfoil, Thin airfoil theory
About this page
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URIm2://NJZJJA2PG8
Kinddefinition
WritingW0 stub
MSC76G25, 76B10
DefinesThin airfoil theory
UsesPotential flow, Circulation, Kutta condition, Kutta–Joukowski theorem, Airfoil equation, Cauchy principal value
Changed3 November 2026

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