Subsonic Aerodynamics | Coefficients | PPL(A) Principles of Flight

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Lift Coefficient (cL)

The lift coefficient (cL) is a dimensionless number that quantifies how effectively a wing generates lift. It depends on:

• Airfoil shape (camber and thickness)
• Angle of attack (AoA)
• Aerodynamic conditions

Changing cL allows optimisation of lift for different flight conditions. Lift increases linearly with cL — a proportional change in cL produces a proportional change in lift.

Lift Formula

L = ½ × ρ × V² × S × cL

Symbol	Meaning
L	Lift force (Newtons)
ρ	Air density (kg/m³)
V	Airspeed (m/s)
S	Wing surface area (m²)
cL	Lift coefficient (dimensionless)

Lift depends on cL, dynamic pressure (½ρV²), and wing area.

Worked Example — Lift

L = cL × ½ × ρ × V² × S

Example: cL = 1, ρ = 1.225 kg/m³, V = 10 m/s, S = 20 m²

L = 1 × ½ × 1.225 × 10² × 20 = 1225 N

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cL vs Angle of Attack

The cL vs AoA graph shows a consistent shape across all airfoil types:

• At low to moderate AoA, cL increases roughly linearly with angle of attack
• Each airfoil has its own zero-lift angle, influenced by camber
• Beyond the critical angle of attack, stall occurs — cL drops sharply

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cL by Airfoil Type at 0° AoA

Airfoil type	cL at 0° AoA	Behaviour with increasing AoA
Symmetrical	0	Increases linearly from zero
Positively cambered	Positive	Starts above zero, increases further
Negatively cambered	Negative	Starts below zero, rises toward positive

Symmetrical Wing at 0° AoA

At 0° AoA, a symmetrical airfoil’s chord is parallel to the airflow. Upper and lower surfaces experience equal pressure — lift forces balance and net lift is zero. This makes symmetrical airfoils ideal for aerobatics and roles requiring neutral lift.

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Drag Coefficient (CD)

Aerodynamic drag is calculated using the same general form as lift:

D = ½ × ρ × V² × S × CD

Symbol	Meaning
D	Drag force (Newtons)
CD	Drag coefficient (dimensionless)
ρ	Air density (kg/m³)
V	Airspeed (m/s)
S	Surface area generating drag (wing, fuselage, etc.)

CD depends on shape, orientation, and surface finish. Drag force combines CD, dynamic pressure, and surface area.

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Induced Drag Coefficient (CDi)

Induced drag is a by-product of lift generation. Its coefficient is given by:

CDi = cL² ÷ (π × AR)

Symbol	Meaning
CDi	Induced drag coefficient
cL	Lift coefficient
AR	Wing aspect ratio

Key relationships:

• Higher AR → lower induced drag
• Higher AoA → higher cL → higher induced drag
• Higher weight → higher AoA required → higher induced drag
• Higher speed → lower AoA required → lower induced drag
• Winglets reduce induced drag by limiting tip vortex formation

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Parasite Drag

Parasite drag acts on all surfaces regardless of lift production:

DP = ½ × ρ × V² × S × CDP

Symbol	Meaning
DP	Parasite drag (Newtons)
CDP	Parasite drag coefficient
ρ	Air density (kg/m³)
V	Airspeed (m/s)
S	Wing surface area (m²)

Worked Examples — Parasite Drag

Effect of doubling airspeed (V² relationship — drag increases by factor of 4):

DP = 1800 N → DPnew = 4 × 1800 = 7200 N

Effect of doubling wing area (linear relationship — drag doubles):

DP = 1800 N → DPnew = 2 × 1800 = 3600 N

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Centre of Pressure (CP)

The centre of pressure is the point along the chord line where the resultant aerodynamic force acts — where the pressure distribution produces no pitching moment.

• For subsonic flow below Mach 0.4, CP is typically at 25% chord (the quarter-chord point), regardless of camber, thickness, or AoA
• CP lies on the chord line

CP Behaviour by Airfoil Type

Airfoil type	CP behaviour
Symmetrical	Remains nearly fixed near quarter-chord for all AoA — aids predictable, stable aerodynamic behaviour
Cambered	Shifts forward as AoA increases; shifts aft as AoA decreases

Why CP moves on cambered airfoils: As AoA increases, lift concentrates nearer the leading edge — CP moves forward. As AoA decreases, lift concentration moves rearward — CP shifts aft toward the trailing edge.

CP and the Transition Point

For a cambered wing, increasing AoA (below critical) moves both the centre of pressure and the laminar-to-turbulent transition point forward. Both reverse when AoA decreases.

Symmetrical Airfoil Applications

Because the CP remains nearly fixed on a symmetrical airfoil, these are common on aerobatic aircraft, training gliders, and UAVs — where predictable, stable aerodynamic behaviour is essential.

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Airfoil Behaviour by Camber

Symmetrical Airfoil

• Identical upper and lower surfaces — no camber
• At 0° AoA: airflow is symmetrical over both surfaces → no net lift
• At positive AoA: airflow over the top surface travels farther → higher speed → lower pressure; bottom surface airflow is slower → higher pressure → upward lift
• At negative AoA: air travels a shorter path over the top surface → pressure increases on top, decreases on bottom → negative lift (downward)

Positively Cambered Airfoil

• More curvature on the upper surface, less on the lower
• Produces positive lift at 0° AoA — lift increases further with AoA
• Zero-lift AoA is at a negative angle of attack
• At zero-lift AoA: suction on upper surface equals suction on lower surface → no net vertical force

Negatively Cambered Airfoil

• Higher bottom surface, lower top surface — creates a downward lift tendency
• At 0° AoA: produces negative lift
• Zero lift occurs at a positive angle of attack, where the camber-induced downward force balances the AoA-induced upward lift

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Critical Angle of Attack and Stall

Some airflow separation occurs at all angles of attack. As AoA increases:

• Separation over the upper surface grows progressively
• At the critical angle of attack: separation becomes large — lift reduces sharply, drag increases significantly
• Beyond the critical angle: strong separation causes major disruption to airflow and loss of lift efficiency — this is the stall

Stall — Key Point

Stall is always caused by exceeding the critical angle of attack — not by low speed alone. Any airfoil can stall at any speed if the critical AoA is exceeded.