Aerodynamics: potential flow, airfoil and wing theory, boundary layers and compressible flow
Aerodynamics tests the chain from inviscid potential flow through finite-wing theory to boundary layers and compressible flow - the section where dimensional errors and thin-airfoil assumptions are most often punished.
Core rules
- Potential flow: continuity gives ∇²φ = 0; u = ∂φ/∂x; the stream function satisfies u = ∂ψ/∂y, v = −∂ψ/∂x. Bernoulli p + ½ρV² = constant along a streamline.
- Thin airfoil theory: lift slope 2π per radian (≈0.11 per degree); lift acts at the quarter chord; camber shifts the zero-lift angle to α_L=0 = −(1/π)∫(dz/dx)(cos θ − 1)dθ.
- Finite wing: downwash reduces the slope to a = a0/(1 + a0/(πeAR)); induced drag CDi = CL²/(πeAR); elliptic loading minimises it.
- Boundary layers: Blasius laminar thickness grows as x^0.5, turbulent as x^0.8; separation occurs under an adverse pressure gradient.
- Compressible flow: stagnation ratios T0/T = 1 + (γ−1)M²/2; choking at M = 1 in a throat; normal shocks raise pressure and entropy while dropping total pressure.
Worked example
For AR = 8, e = 0.9, CL = 0.6: CDi = 0.6²/(π × 0.9 × 8) = 0.36/22.62 = 0.0159. The lift slope becomes 2π/(1 + 2/(0.9×8)) ≈ 4.92 per radian - visibly below the two-dimensional 6.28.
Traps
- Mixing degrees and radians in lift-slope relations; 2π per radian is 0.11 per degree.
- Using static pressure where stagnation pressure is required in isentropic relations.
- Assuming a normal shock conserves total pressure (it does not) or total temperature (it does).
- Forgetting Prandtl–Glauert for compressibility effects, and misapplying it beyond the critical Mach number.