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1. Small-Disturbance Flow over 3D Wings Based on Chapter 4 of Katz and Plotkin's 'Low-Speed Aerodynamics'

2. Introduction • Study of lifting surfaces using potential flow theory • Simplifying assumptions to handle boundary conditions on complex shapes • Application of singularity methods to solve 3D wing problems

3. Definition of the Problem • Wing in a uniform free-stream • Coordinate system: x (chordwise), y (spanwise), z (vertical) • Assumptions: Inviscid, incompressible, irrotational flow • Governing equation: Laplace equation for velocity potential ∇²ϕ* = 0 • Boundary conditions: - Disturbance decays at infinity - No normal flow on wing surface

4. Boundary Condition on the Wing • Wing surface equation: z = η(x, y) • Normal to wing: n = (-∂η/∂x, -∂η/∂y, 1) • Perturbation potential ϕ* = ϕ + ϕ∞ • Simplified boundary condition: ∂ϕ/∂z = U∞ (∂η/∂x - α) at z = η

5. Small-Disturbance Approximation • Assumes small perturbations in velocity and angle of attack: |∂ϕ/∂x|/Q∞, |∂ϕ/∂y|/Q∞, |∂ϕ/∂z|/Q∞ << 1 • Wing must be thin (∂η/∂x, ∂η/∂y << 1) • Boundary condition transferred to z = 0 plane: ∂ϕ/∂z (x, y, 0) = Q∞ (∂η/∂x - α)

6. Separation of Thickness and Lifting Problems • Wing shape decomposed into: 1. Symmetric wing (thickness effect) 2. Cambered wing (lifting effect) 3. Wing at an angle of attack • Each problem solved separately and results superimposed

7. Symmetric Wing Thickness Effect • Modeled using a source distribution: σ(x, y) = 2Q∞ ∂ηt/∂x • Velocity field computed using potential theory • No lift is generated due to thickness alone

8. Lifting Problem with Angle of Attack • Boundary condition at z = 0 plane: ∂ϕ/∂z = Q∞ (∂ηc/∂x - α) • Modeled using: - Doublet distribution - Vortex sheet distribution • Wake must be introduced to satisfy Helmholtz's theorem

9. Doublet Distribution for Lifting Surfaces • Velocity potential: ϕ = -1/(4π) ∫ μ(x0, y0) z / [(x-x0)² + (y-y0)² + z²]³/² dx0 dy0 • Induced velocity components: - u = -1/2 ∂μ/∂x - v = -1/2 ∂μ/∂y - w obtained by integral equation

10. Vortex Distribution for Lifting Surfaces • Uses Biot-Savart law to compute induced velocity: q = -1/(4π) ∫ (Γ × dl) / r³ dx0 dy0 • Wake modeled as vortex sheet • Enforces downwash equation for lift computation

11. Aerodynamic Loads • Computed using Bernoulli equation: p∞ - p = ρQ∞ ∂ϕ/∂x • Lift force: L = ∫ (p_l - p_u) dx dy • Coefficients: C_L = L / (0.5 ρ Q∞² S) C_M = M / (0.5 ρ Q∞² S b)

12. Vortex Wake Considerations • Wake must be parallel to free-stream: Q∞ × γw = 0 • Wake carries vorticity shed from wing • Satisfies Kutta condition at trailing edge