Bernoulli's Equation Flow Calculator

Bernoulli’s equation expresses conservation of energy for steady, incompressible, inviscid fluid flow:

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

Where:

• P = Static pressure (Pa)
• ρ = Fluid density (kg/m³)
• v = Flow velocity (m/s)
• g = Gravitational acceleration = 9.81 m/s²
• h = Height above reference level (m)

Dynamic pressure: q = ½ρv² (increases as speed increases) Static pressure: P (decreases as speed increases) Total pressure: P₀ = P + ½ρv² (constant along a streamline)

Real-world applications:

• Airplane wings: Curved upper surface → faster air → lower pressure → lift force
• Carburetors: Fast-moving air through venturi creates low pressure to draw in fuel
• Water towers: High elevation provides pressure in pipes below
• Pitot tubes: Measure aircraft airspeed by comparing static and stagnation pressures
• Garden hoses: Covering part of the nozzle increases velocity (continuity) and decreases pressure

Assumptions and limitations: Bernoulli’s equation assumes:

1. Steady flow (no turbulence)
2. Incompressible fluid (good for liquids, valid for air below Mach 0.3)
3. No viscosity (friction) along the streamline
4. No work done by pumps or extracted by turbines

Where that bites in practice. The no-viscosity assumption is the one that fails first. Run water down a long pipe and pressure drops even where the diameter never changes, because friction at the wall is dissipating energy that Bernoulli’s equation says should still be there. For pipe runs of any length you need the Darcy-Weisbach head-loss term on top of this. Bernoulli is exact for a short converging nozzle and increasingly optimistic for a hundred metres of pipe.

The second thing worth knowing is that the equation only holds along one streamline. Comparing a point above a wing to a point below it is comparing two different streamlines, and that is precisely why the popular “equal transit time” story about lift is wrong: nothing requires the air over the top to rejoin the air underneath, and measurements show it arrives earlier, not at the same time. The pressure difference over a wing is real and Bernoulli describes it correctly once you have the velocity field, but the velocity field comes from circulation and the shape of the flow, not from the two paths being the same length.

A worked check you can do at a tap. Water leaving an open tank under gravity alone reduces to Torricelli’s result, v = √(2gh), because both pressures are atmospheric and the velocity at the top surface is negligible. A tank with 2 m of water above the outlet gives √(2 × 9.81 × 2) = 6.3 m/s. That is a special case of the equation above, not a separate rule.

For real pipes, add a friction head loss term or use the Darcy-Weisbach equation.