Subsonic Aerodynamics | 2D Flow Around Airfoils | PPL(A) Principles of Flight

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Streamlines

A streamline is a line tangent to the local velocity vector at every point in the flow. In steady flow, streamlines, pathlines, and streaklines are all the same — they can be visualised by particles moving with the flow.

What streamlines tell us:

• Flow direction — streamlines show the direction of airflow at every point in the field
• Relative velocity — the spacing between streamlines indicates local flow speed
• Flow separation — a sudden change in streamline direction indicates detachment from the surface

Streamline spacing and pressure (Bernoulli’s principle):

Streamline spacing	Velocity	Static pressure
Closer together	Higher	Lower
Further apart	Lower	Higher

In incompressible flow, this relationship is exact: as the flow accelerates through a narrowing stream tube, static pressure drops proportionally.

Streamlines and Bernoulli

Closer streamlines → higher velocity → lower static pressure.
Wider streamlines → lower velocity → higher static pressure.
This is a direct consequence of Bernoulli’s theorem and the continuity equation.

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Stagnation Point

The stagnation point is the location on the airfoil surface where the oncoming airflow divides — one stream going over the upper surface, the other under the lower surface. At this point, the local flow velocity drops to zero.

Because all kinetic energy converts to pressure energy at this point:

Total pressure = Static pressure + Dynamic pressure

This is the maximum pressure point on the airfoil. On a wing, the stagnation point sits near the leading edge, slightly below the chord line, due to the asymmetric splitting of airflow around the wing.

How this is used for airspeed measurement:

• A Pitot tube faces the airflow and captures total (stagnation) pressure
• Static ports measure undisturbed static pressure parallel to the airflow
• The difference gives dynamic pressure: Dynamic pressure = Total pressure − Static pressure
• From dynamic pressure, airspeed can be calculated for the airspeed indicator

Stagnation Pressure Formula

At the stagnation point: P_total = P_static + ½ρV²

Rearranging: ½ρV² = P_total − P_static

This dynamic pressure value is what the airspeed indicator uses to derive airspeed.

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Pitch, Angle of Attack, and Lift

Pitch refers to raising or lowering the aircraft’s nose using the control yoke or stick. Pitch directly controls the angle of attack (AoA) — the angle between the wing’s chord line and the undisturbed (freestream) airflow.

Lift is calculated as:

L = ½ × ρ × V² × S × cL

The lift coefficient (cL) rises with increasing angle of attack — so pitch is the primary in-flight control over lift production.

Pitching Up — Increasing AoA

• Increases the angle of attack
• Raises the lift coefficient (cL) and total lift production
• Beyond the critical angle of attack, airflow separation occurs over the upper surface
• cL drops sharply — lift decreases — this is the stall
• Effective pitch-up control balances lift needs with stall avoidance

Pitching Down — Decreasing AoA

• Decreases the angle of attack
• Reduces the lift coefficient (cL) and total lift
• Recovers from a stall by reducing AoA below the critical angle
• Effective pitch-down control ensures proper performance management across all flight phases

Critical Angle of Attack

Beyond the critical angle of attack, airflow becomes turbulent and separates from the upper surface. cL drops sharply and lift decreases significantly — this is the stall.
Stall is caused by exceeding the critical AoA — not by low speed alone.

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Airfoil Types and Camber

Camber is the curvature of the mean camber line relative to the chord line. It determines the lift characteristics at any given angle of attack — including where the zero-lift angle falls.

Symmetrical Airfoil

A symmetrical airfoil has identical upper and lower surfaces — no camber. The mean camber line coincides with the chord line.

• At 0° AoA: the chord line is parallel to the airflow; airflow and pressure distribution are equal on both surfaces — lift forces balance and net lift is zero
• 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)
• The zero-lift angle is exactly 0° AoA

Symmetrical Airfoil Applications

Because a symmetrical airfoil produces zero lift at 0° AoA and behaves predictably at both positive and negative angles, it is commonly used on aerobatic aircraft, training gliders, and UAVs — where neutral, stable aerodynamic behaviour is essential.

Positively Cambered Airfoil

A positively cambered airfoil has more curvature on the upper surface and a flatter lower surface, enhancing lift generation.

• At 0° AoA: the upper surface curvature still accelerates airflow → positive lift is produced even at zero angle of attack
• Lift increases further as AoA increases
• The zero-lift angle is at a negative angle of attack — the wing must be tilted slightly nose-down to produce no lift
• At the zero-lift AoA: suction on the upper surface equals suction on the lower surface → no net vertical force
• Camber shifts the zero-lift point below the chord line, requiring a slight nose-down orientation to produce no lift

Negatively Cambered Airfoil

A negatively cambered airfoil has a higher bottom surface and lower top surface, creating a natural downward lift tendency.

• At 0° AoA: produces negative lift (downward force)
• The zero-lift angle is at a positive angle of attack — the camber-induced downward force must be balanced by the AoA-induced upward lift to achieve zero net lift
• Used in specific applications where a download is required (e.g. inverted flight stabilisers, some tail surfaces)

Summary — Zero-Lift Angle by Camber Type

Airfoil type	cL at 0° AoA	Zero-lift AoA
Symmetrical	0	0°
Positively cambered	Positive	Negative (nose-down)
Negatively cambered	Negative	Positive (nose-up)

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Centre of Pressure and the Transition Point

Centre of Pressure (CP)

Lift acts across the entire wing surface but can be represented as concentrated at a single point — the centre of pressure (CP). This is the point along the chord line where the resultant aerodynamic force acts, producing no net pitching moment.

The CP’s position is not fixed — it moves with angle of attack and airfoil shape:

• Increasing AoA moves the CP forward — upper-surface suction shifts toward the leading edge as lift develops nearer the front of the wing
• Forward CP movement increases lift until the stall is approached
• Just before stall, the CP is at its maximum forward position (corresponding to cL MAX)
• At stall, lift drops sharply and the CP shifts rearward
• Decreasing AoA moves the CP aft toward the trailing edge

CP Behaviour by Airfoil Type

Symmetrical airfoil: CP remains nearly fixed near the quarter-chord point (25% chord) for all angles of attack — this gives predictable, stable aerodynamic behaviour.

Cambered airfoil: CP shifts forward as AoA increases; shifts aft as AoA decreases. This movement must be accounted for in aircraft stability and trim design.

Transition Point

The transition point is where the boundary layer changes from laminar (smooth, layered) flow to turbulent flow along the upper surface.

For a cambered wing, increasing AoA (below the critical angle) moves both the centre of pressure and the transition point forward toward the leading edge:

• As AoA increases → suction peak moves forward → transition point advances toward the leading edge
• As AoA decreases → both CP and transition point move aft

CP and Transition Point — Together

For a cambered wing below the critical AoA:

• Higher AoA → CP moves forward, transition point moves forward
• Lower AoA → CP moves aft, transition point moves aft
  Both effects reverse simultaneously with changes in angle of attack.