Fundamentals7 min read2 / 9

Bernoulli's principle

Bernoulli’s principle links speed and pressure in a moving fluid: speed up, and pressure drops. It is the bridge between the “flow” picture and the “pressure” picture — but it is also the single most mis-taught idea in flight, so we will state exactly what it does and does not say.

The equation

For a steady, incompressible, inviscid flow, the sum of pressure, kinetic energy per unit volume, and gravitational potential per unit volume is constant along a streamline:

p + \tfrac12\rho V^2 + \rho g z = \text{constant}

(1)

$p$: static pressure [Pa]
$\tfrac12\rho V^2$: dynamic pressure (kinetic energy per unit volume) [Pa]
$\rho g z$: hydrostatic term (height z) [Pa]

For air over a wing the height term is negligible, so we use the compact form between any two points on a streamline:

p_1 + \tfrac12\rho V_1^2 = p_2 + \tfrac12\rho V_2^2

(2)

Where it comes from

Apply Newton’s second law to a fluid element sliding along a streamline (Euler’s equation in the flow direction $s$ ):

\rho V\,\frac{dV}{ds} = -\frac{dp}{ds}

(3)

The left side is mass × acceleration per unit volume; the right side is the net pressure force per unit volume. Recognising $V\,dV = d(\tfrac12 V^2)$ and integrating along the streamline gives $\tfrac12\rho V^2 + p = \text{const}$ — equation (1) without gravity. No friction appears because we assumed an inviscid fluid.

Fig. 1. Squeeze a streamtube and the same mass of air must move faster (continuity). Faster flow means lower pressure (Bernoulli).

Why the air speeds up over the top

Two ideas combine. Continuity (conservation of mass) says that in a steady flow the same mass crosses every section of a streamtube each second, so where streamlines bunch together the air must speed up. Over the curved upper surface the streamlines are squeezed, so the flow accelerates; by (2) its pressure falls. That low pressure on top is most of the lift.

When Bernoulli is allowed

Equation (1) assumes the flow is steady, incompressible (density essentially constant — true for air below about Mach 0.3, i.e. ~100 m/s) and inviscid (no friction). It holds along a single streamline; if the flow is also irrotational — as ideal flow around a wing is — the constant is the same for every streamline, so we may compare any two points in the field. Inside the thin boundary layer next to the skin, viscosity matters and Bernoulli no longer applies.