Bernoulli's principle

Bernoulli's Principle

Bernoulli's principle describes energy conservation in inviscid, incompressible, steady flow along a streamline. Daniel Bernoulli formulated it in his Hydrodynamica (1738).

Equation

Along a streamline:

p + ½ ρ v² + ρ g h = constant

with:

• p = static pressure
• ½ρv² = dynamic pressure (kinetic term)
• ρgh = gravitational pressure (height term)

In horizontal flow (gh ≈ const, drops out):

p + ½ ρ v² = pt = constant

Statement

• Where flow is faster (higher v), static pressure p is lower.
• Where flow is slower, p is higher.
• At rest (v=0, e.g. at a stagnation point) p reaches its maximum value pt.

Application to a wing

Above the curved upper surface the flow accelerates → lower p (pressure on top lower than ambient). Below the flatter underside it flows slower → higher p. Pressure difference, lower vs upper → lift.

Note — common false explanation: The popular explanation "the air on top must be faster so it arrives at the trailing edge at the same time as the air below" is wrong. The flow on top reaches the trailing edge faster than the one below — there is no need for them to arrive simultaneously. Correct explanations rely on circulation and the Kutta condition (see lesson "Lift generation").

Validity limits

Bernoulli applies only under:

1. Incompressible: Mach number M < 0.3 (≈ 100 m/s = 200 kt). PPL speeds ✓.
2. Inviscid: boundary layer effects not modelled. Real wing surface has a boundary layer → small correction.
3. Steady: time-invariant flow.
4. Along a streamline: NOT between different streamlines (except in special forms).

Venturi effect

Application: at a constriction of a flow channel the flow accelerates → p drops. Used in:

• Carburettor (for fuel intake — see Subject 050 lesson "Engine / Induction Icing").
• Venturi tube for vacuum generation (Subject 020 — vacuum system for gyros).
• Wingtip vortex and streamline convergence.

Pitot-static application

Airspeed indicator (ASI) measures pt − p = ½ρ₀v², calibrated to MSL density ρ₀. From it IAS (indicated airspeed). For TAS (true airspeed), correct with current ρ.