How does an aerofoil generate lift?

An aerofoil generates lift by forcing air to travel faster over its upper (cambered/angled) surface than beneath. By Bernoulli's principle, higher velocity means lower pressure on top, creating an upward pressure difference. The Kutta-Joukowski theorem quantifies this: L = ?V8G per unit span, where G is the bound circulation generated by the aerofoil's angle of attack and camber.

What is a NACA aerofoil?

NACA (National Advisory Committee for Aeronautics) aerofoils are standardised wing cross-sections defined by 4-digit codes. NACA XYZZ: X = maximum camber as % chord, Y = chordwise position of max camber in tenths, ZZ = maximum thickness as % chord. E.g. NACA 2412 has 2% camber at 40% chord and 12% thickness. The geometry is generated by analytical equations using cosine spacing for accurate panel coverage.

What happens at stall?

At angles of attack above ~14–16°, the boundary layer on the upper surface separates from the aerofoil, causing a dramatic loss of lift and increase in pressure drag — a stall. The simulator shows the ? STALL warning when a exceeds 14°, and CD rises sharply while CL plateaus then drops. In real aircraft, stall recovery requires reducing angle of attack.

What is the panel method?

The panel method approximates the flow around an aerofoil by distributing vortex panels along its surface. Each panel has an unknown vortex strength ? determined by enforcing zero normal velocity at control points (the Neumann boundary condition) plus the Kutta condition (smooth flow at the trailing edge). Solving this linear system gives the pressure distribution, lift, and the induced velocity field for streamline tracing.

Aerodynamics Simulator – NACA Aerofoil Lift, Drag & Panel Method

Bernoulli's Principle and Lift Generation

Lift arises because an aerofoil accelerates air over its upper surface more than its lower surface. Bernoulli's principle relates flow speed to static pressure along a streamline:

p + ½?v² + ?gh = constant (along a streamline)

Where v is higher (suction side), p is lower. The pressure difference across the chord creates a net upward force. The pressure coefficient Cp non-dimensionalises this:

Cp = (p - p8) / (½?V8²) = 1 - (v/V8)²

On the upper surface Cp is negative (below 0); on the lower surface Cp is positive. The area between the Cp curves in the chart equals the lift coefficient CL:

CL = ? Cp · cos(?) ds/c (integrated around chord)

For thin aerofoils (Kutta-Joukowski): L = ? V8 G per unit span, where G is the total bound circulation.

Thin Aerofoil Theory

For inviscid, incompressible flow, thin-aerofoil theory gives a remarkably simple result: the lift-curve slope is 2p per radian regardless of aerofoil shape:

CL ˜ 2p (a + 2m) [rad, for small a and camber m]
dCL/da = 2p ˜ 0.1097 per degree

Camber m shifts the zero-lift angle (NACA 2412 lifts at a = 0°), while thickness affects Cp distribution but not CL in first-order theory.

Panel Method Explained

The panel method solves potential-flow aerodynamics by distributing vortex panels along the aerofoil surface. At each control point (panel midpoint), the normal velocity must be zero — the flow is tangent to the surface. This yields N linear equations in N unknown vortex strengths γⱼ.

S? Aᵢⱼ γⱼ = -V8 · n^? (i = 1…N) Kutta condition: ?1 + γₙ = 0 (smooth trailing-edge flow)

Solving the linear system (Gaussian elimination) gives ?, the velocity field, Cp, CL and the streamlines. The induced velocity at any point (x,y) from panel j is:

u = -γⱼ(y-yⱼ) / (2p r²) v = γⱼ(x-xⱼ) / (2p r²)

Streamlines via Runge-Kutta

Each streamline starts at the left boundary and follows the velocity field using 4th-order RK integration:

k1 = V(x?, y?) k2 = V(x? + k1?t/2, y? + k1?t/2) ... xγⱼ1 = x? + (k1 + 2k2 + 2k3 + k4)?t/6

NACA 4-Digit Aerofoil Profiles

NACA 4-digit profiles are defined by three parameters encoded in the name NACA MPTT:

NACA Code	Max Camber m	Camber Pos p	Thickness t	Application
NACA 0012	0% (symmetric)	—	12%	Tail surfaces, wind turbines
NACA 2412	2% at 40%	40%	12%	General aviation, Cessna 172
NACA 2415	2% at 40%	40%	15%	High-lift training aircraft
NACA 4412	4% at 40%	40%	12%	Light aircraft, paragliders
NACA 0006	0% (symmetric)	—	6%	Supersonic fins (thin)

The thickness distribution uses the NACA standard formula with cosine spacing for accuracy:

y?(x) = (t/0.2)[0.2969vx - 0.1260x - 0.3516x² + 0.2843x³ - 0.1015x4]

Lift, Drag and Aerodynamic Efficiency

Parameter	Symbol	Typical Values	Notes
Lift coefficient	CL	0 – 1.8 (pre-stall)	Increases linearly with a at 2p/rad
Drag coefficient	CD	0.006 – 1.2	Parasitic + induced; rises sharply at stall
Lift-to-drag ratio	L/D = CL/CD	20–200 for gliders	Aerodynamic efficiency; maximised at optimal a
Reynolds number	Re = V8 c / ?	105 – 108	Low Re ? laminar; high Re ? turbulent BL
Stall angle	a_stall	12° – 18°	Depends on surface roughness and Re
Zero-lift angle	a0	-2° to 0°	Non-zero for cambered aerofoils

Drag Polar

The drag polar relates CD to CL:

CD = CD0 + CL² / (p e AR) (induced drag)

Where CD0 is parasitic drag, e is Oswald efficiency (0.7–0.9 for real wings) and AR is aspect ratio. A high-aspect-ratio wing (sailplane: AR˜30) minimises induced drag.

Reynolds Number and Flow Regimes

Re = ? V L / µ = V L / ?

The Reynolds number governs whether flow is laminar (smooth, layered) or turbulent (chaotic, high mixing). For an aerofoil with chord L = 1 m:

Re range	Flow type	Boundary layer	Application
< 104	Viscous / creeping	Thick laminar	Insects, micro-UAVs
104 – 106	Transitional	Laminar separation bubble	Model aircraft, small drones
106 – 108	Turbulent	Thin turbulent BL	Commercial aircraft, wind turbines
> 108	High-Re turbulent	Fully turbulent	High-speed aircraft in dense air

Turbulent boundary layers resist separation better (higher stall angle, more robust CL) but have higher skin friction drag due to greater momentum transfer at the wall.

Curriculum Connections

Topic	Qualification	Concepts Covered
Fluid mechanics — Bernoulli	A-Level Physics	Continuity equation, Bernoulli, venturi effect, pitot tube
Forces and Newton's 3rd Law	GCSE / A-Level Physics	Lift:reaction force, drag, Newton's laws in fluid context
Numerical methods	A-Level Further Maths	Linear systems (Gaussian elimination), Runge-Kutta ODE
Transport engineering	BTEC Engineering	Aerodynamic efficiency, drag polar, wing design
Computational modelling	A-Level CS / IB	Panel method discretisation, mesh independence
Dimensional analysis	A-Level / University	Reynolds number, non-dimensional CL/CD, similarity

Aerodynamics Simulator