The Bernoulli equation states that along a streamline in steady, inviscid, incompressible flow, the sum of static pressure, dynamic pressure, and hydrostatic pressure is constant:
p + ½ρv² + ρgh = constant
where p is static pressure, ρ is fluid density, v is flow velocity, g is gravitational acceleration, and h is height. In most aerodynamic applications, the height term is negligible:
p + ½ρv² = p₀
where p₀ is the total (stagnation) pressure. The key implication: where velocity increases, pressure decreases.
How lift works (and how it doesn’t)
The Bernoulli equation is commonly invoked to explain lift: air moves faster over the curved upper surface of an airfoil, so pressure is lower above the wing than below, producing a net upward force. This is correct as a description of the pressure distribution but incomplete as an explanation — it does not explain why the air moves faster over the top. The complete explanation requires the Navier-Stokes equations, circulation theory (the Kutta condition), and the role of viscosity in establishing the flow pattern.
The common “equal transit time” explanation — that air molecules must traverse the longer upper path in the same time as the shorter lower path — is simply wrong. Air over the top of a wing arrives at the trailing edge before air flowing under the bottom, not at the same time.
Limitations
The Bernoulli equation is valid only for:
- Steady flow — no time-varying changes
- Incompressible flow — Mach number below ~0.3
- Inviscid flow — outside the boundary layer
- Along a streamline — not across streamlines (unless the flow is irrotational)
Despite these limitations, the Bernoulli equation is remarkably useful for first-order analysis: pitot tubes measure airspeed using it, venturi flow meters depend on it, and it provides good estimates of pressure distributions on airfoils at low Mach numbers.
Related terms
- Dynamic Pressure — the ½ρv² term in the equation
- Navier-Stokes Equations — the full equations from which Bernoulli is derived as a special case
- Lift — the aerodynamic force explained (in part) by Bernoulli’s pressure relationship