Thin airfoil theory

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Thin Airfoil Theory is a fundamental analytical model in aerodynamics that predicts the lift and moment characteristics of thin, cambered airfoils in inviscid, incompressible, 2D flow.

It provides relatively simple but remarkably accurate approximations for engineering use, especially at small angles of attack.

1️⃣ Assumptions of Thin Airfoil Theory

  • Airfoil is thin (small thickness-to-chord ratio).
  • Small angle of attack (linear approximation).
  • Flow is inviscid and incompressible.
  • Flow is steady and 2D.
  • The Kutta condition applies at the trailing edge (smooth flow leaving TE).

✅ These simplifications make the mathematics tractable while capturing key aerodynamic behavior.

2️⃣ Conceptual Approach

Thin Airfoil Theory models the airfoil surface with a distribution of bound vortices along the chord line.

Key idea:

Replace the real airfoil with its camber line plus a vortex sheet of varying strength.

✅ The strength of the vortex distribution is chosen so that the flow tangency condition (no normal flow through the camber line) is satisfied.

3️⃣ Mathematical Formulation

The fundamental integral equation for the camber line is:

\alpha - \frac{dz}{dx} = \frac{1}{2\pi} \int_{0}^{c} \frac{\gamma(\xi)}{x - \xi} d\xi

Where:

  • \alpha: angle of attack (in radians).
  • \frac{dz}{dx}: local slope of the camber line.
  • \gamma(\xi): vortex strength per unit length at position ξ\xi.

✅ This is the flow tangency condition along the camber line.

4️⃣ Solution for Symmetric Airfoil

For a symmetric airfoil (camber = 0):

\frac{dz}{dx} = 0

The integral equation reduces to:

\alpha = \frac{1}{2\pi} \int_{0}^{c} \frac{\gamma(\xi)}{x - \xi} d\xi

✅ Solution yields a linear lift curve with zero-lift angle of attack at 0°.

5️⃣ General Cambered Airfoil Solution

For a cambered airfoil, the integral equation includes the camber slope:

\alpha - \frac{dz}{dx} \neq 0

✅ Result: the airfoil develops lift at zero geometric angle of attack.
✅ The zero-lift angle of attack \alpha_0 becomes negative.

6️⃣ Lift Coefficient from Thin Airfoil Theory

The famous result:

C_L = 2\pi (\alpha - \alpha_0)

Where:

  • α: angle of attack in radians.
  • α0: zero-lift angle of attack.

✅ The lift curve slope is 2π per radian.
✅ Very good approximation for many real airfoils at low angles of attack.

7️⃣ Zero-Lift Angle of Attack

Calculated from camber line shape:

\alpha_0 = - \frac{1}{\pi} \int_{0}^{\pi} \frac{dz}{dx} d\theta

Where:

  • θ: transformed variable along the chord (via mapping).
  • \frac{dz}{dx}: camber line slope function.

✅ Cambered airfoils have negative α0.

8️⃣ Pitching Moment Coefficient

Thin Airfoil Theory also predicts pitching moment about the quarter-chord point:

C_{m,c/4} = \text{constant (independent of } \alpha \text{)}

✅ For symmetric airfoils: C_{m,c/4} = 0.
✅ For cambered airfoils: C_{m,c/4} is typically negative (nose-down moment).
✅ Crucial for longitudinal stability calculations.

9️⃣ Example Application

Problem:
Find the lift coefficient at α=5° (≈0.087 radians) for a symmetric airfoil.

Solution:

  • α0 = 0 (symmetric).
  • C_L = 2\pi \alpha = 2\pi \times 0.087 \approx 0.55

✅ Quick, accurate estimate of lift at small angles of attack.

1️⃣0️⃣ Limitations of Thin Airfoil Theory

  • Ignores thickness effects (pressure distribution on real, thick airfoils).
  • Assumes inviscid flow (no boundary layer, no separation).
  • Valid only for small angles of attack (linear range).
  • 2D approximation (no 3D effects like spanwise flow or induced drag).

✅ Despite these, it remains a cornerstone of classical aerodynamics.

Summary

✅ Thin Airfoil Theory provides analytical predictions of lift and moment for thin, cambered airfoils.
✅ Predicts linear lift curve with slope 2π per radian.
✅ Accounts for camber through zero-lift angle of attack and pitching moment.
✅ Simple, powerful, and foundational for understanding airfoil behavior.

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