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Thin Airfoil Theory

Field: Aerodynamics

Sequence of Expressions

Let Ω\Omega be the flow domain and S\mathcal{S} be the airfoil surface. The assumptions impose the following constraints on the governing equations: (1) Incompressibility: ∇⋅v⃗=0\nabla \cdot \vec{v} = 0 in Ω\Omega. (2) Inviscid Flow: μ=0\mu = 0 (viscosity). (3) Thin Airfoil Limit: The maximum thickness tt and the chord length cc satisfy t/c≪1t/c \ll 1, allowing the flow to be modeled by a linearized boundary condition on the chord line.
Let zTz_T be the trailing edge point. The Kutta condition requires the velocity field v⃗\vec{v} to be smooth at zTz_T, implying that the jump in the normal component of the velocity across the trailing edge must vanish: lim⁡ϵ→0[v⃗⋅n⃗]upper→lower=0\lim_{\epsilon \to 0} [\vec{v} \cdot \vec{n}]_{\text{upper} \to \text{lower}} = 0. This condition determines the unique circulation Γ\Gamma such that the flow leaves the trailing edge tangentially.
Define the circulation Γ\Gamma around a closed contour CC enclosing the airfoil cross-section C\mathcal{C} as the line integral of the tangential component of the velocity field v⃗\vec{v}: Γ=∮Cv⃗⋅dl⃗=Im(∮Cdz⃗z−z0)\Gamma = \oint_C \vec{v} \cdot d\vec{l} = \text{Im} \left( \oint_C \frac{d\vec{z}}{z-z_0} \right) where z0z_0 is a point outside CC.
Define the lift coefficient ClC_l based on the angle of attack α\alpha (in radians) using the linearized relationship derived from the Kutta-Joukowski theorem: Cl=2π(α−α0)C_l = 2\pi (\alpha - \alpha_0) where α0\alpha_0 is the zero-lift angle of attack.
Define the angle of attack α\alpha as the angle between the freestream velocity vector U⃗∞\vec{U}_{\infty} and the chord line c⃗\vec{c} of the airfoil. Mathematically, if U⃗∞\vec{U}_{\infty} has direction θ∞\theta_{\infty} and the chord line is aligned with the x-axis (θc=0\theta_c = 0), then α=θ∞\alpha = \theta_{\infty}. The relative velocity v⃗rel\vec{v}_{rel} is given by v⃗rel=U⃗∞−v⃗induced\vec{v}_{rel} = \vec{U}_{\infty} - \vec{v}_{induced}.
The lift force per unit span L′L' is governed by the Kutta-Joukowski theorem: L′=ρU∞ΓL' = \rho U_{\infty} \Gamma For a 2D flow over a chord cc, the lift force L′L' is approximated by: L′≈12ρU∞2c⋅2παL' \approx \frac{1}{2} \rho U_{\infty}^2 c \cdot 2\pi \alpha where ρ\rho is the fluid density and U∞U_{\infty} is the freestream velocity.
Model the fluid flow using a velocity potential ϕ\phi such that the velocity field v⃗\vec{v} is derived by the gradient: v⃗=∇ϕ\vec{v} = \nabla \phi. The governing equation for ϕ\phi in an incompressible, inviscid flow is the Laplace equation: ∇2ϕ=0\nabla^2 \phi = 0 in the flow domain Ω\Omega.
The no-penetration boundary condition requires that the fluid velocity v⃗\vec{v} must be tangential to the airfoil surface S\mathcal{S} at all points. If n⃗\vec{n} is the unit normal vector to S\mathcal{S}, the condition is: v⃗⋅n⃗=0on S\vec{v} \cdot \vec{n} = 0 \quad \text{on } \mathcal{S}
In the complex plane z=x+iyz = x + iy, the complex potential W(z)W(z) for the 2D flow is defined. The governing equation relating W(z)W(z) to the circulation Γ\Gamma and the freestream velocity U∞U_{\infty} is: W(z)=U∞z+12πi∫CΓ(z′)z−z′dz′W(z) = U_{\infty} z + \frac{1}{2\pi i} \int_C \frac{\Gamma(z')}{z-z'} dz' where CC is the contour enclosing the airfoil.
Discretize the airfoil surface S\mathcal{S} into NN panels. The boundary condition v⃗⋅n⃗=0\vec{v} \cdot \vec{n} = 0 is enforced by solving a system of linear equations for the unknown coefficients (e.g., source/dipole strengths) x\boldsymbol{x}: Ax=b\mathbf{A} \boldsymbol{x} = \mathbf{b} where A\mathbf{A} is the influence matrix derived from the panel geometry and b\mathbf{b} is the prescribed boundary condition vector.