Bernoulli Equation & Fluid Flow Calculator

Bernoulli Equation Guide

What's the key thing to understand about Bernoulli's Equation?

P + ½ρv² + ρgh = constant. P = static pressure (Pa), ½ρv² = dynamic pressure (Pa), ρgh = hydrostatic pressure (Pa). Between two points in steady, inviscid, incompressible flow: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. Key insight: as velocity increases, pressure decreases (and vice versa). Example: pipe narrows from 5cm to 2.5cm diameter. Continuity: A₁v₁ = A₂v₂. Flow at 2 m/s through large section: v₂ = v₁ × (A₁/A₂) = 2 × 4 = 8 m/s. Pressure drop: ΔP = ½ρ(v₂² - v₁²) = ½×1000×(64-4) = 30,000 Pa.

What do I need to know about Torricelli's Theorem?

A special case of Bernoulli: the speed of fluid exiting a small hole in a large tank. v_exit = √(2gh). Where h = height of fluid above the exit hole. Same formula as free-fall velocity — the fluid accelerates as if falling from height h. Example: tank with 5m of water above exit: v = √(2×9.81×5) = √98.1 = 9.9 m/s. This approximation assumes: the exit hole is small compared to the tank. The tank level drops slowly. Viscosity is negligible (reasonable for water through large holes). Real orifices lose some energy to turbulence and viscosity, so the actual exit velocity is typically 95-98% of the theoretical Torricelli value.

What's the key thing to understand about Venturi Meter?

A venturi meter measures flow rate by creating a controlled narrowing. Mass flow rate: Q = A₁A₂√(2ΔP/(ρ(A₁²-A₂²))). Pressure drop ΔP = P₁ - P₂ measured by a manometer or differential pressure sensor. Applications: water supply measurement, natural gas metering, carburettors (the venturi principle draws fuel into the airstream). Aircraft pitot-static tube: measures dynamic pressure ½ρv² from which airspeed is calculated. Pitot tube: total pressure = static + dynamic. Static port: static pressure only, measured perpendicular to the airflow so the moving air doesn't directly hit the sensor.

What's the key thing to understand about Bernoulli and Lift?

Aircraft wings (aerofoils) generate lift through a pressure difference — but the mechanism is more nuanced than the textbook explanation. Common (oversimplified) explanation: air travels faster over the curved top surface → lower pressure → upward lift (Bernoulli). More accurate: lift arises primarily from circulation of airflow around the wing (Kutta-Joukowski theorem). The wing deflects air downward — by Newton's 3rd law, the wing is pushed upward. Bernoulli pressure difference is a real effect contributing to lift, but the popular explanation that faster airflow over the curved top surface alone explains flight is an oversimplification — Newton's third law and the wing's angle of attack matter at least as much.

What do I need to know about the Continuity Equation?

Before Bernoulli's equation can be applied, conservation of mass for an incompressible fluid requires A₁v₁ = A₂v₂ — the continuity equation. It says that wherever a pipe narrows, the fluid must speed up to keep the same volume flowing past every cross-section each second. This is precisely why Bernoulli's equation predicts lower pressure in the narrow part of a Venturi tube: continuity forces the speed up there, and Bernoulli then forces the pressure down to compensate, since the sum P + ½ρv² + ρgh must stay constant along the streamline.

What should I know about the Limitations of Bernoulli's Equation?

Bernoulli's equation is an idealisation: it strictly only holds for steady, incompressible, non-viscous flow along a single streamline, with no energy added or removed by a pump, turbine, or friction. Real pipe flow loses energy to viscous friction against the walls (which is why long pipelines need booster pumps), and gases at high speed become compressible, invalidating the constant-density assumption. Engineers handle this by adding a head-loss term for friction and switching to compressible-flow equations once Mach numbers approach about 0.3 — Bernoulli alone is a good first approximation, not the final word, for real engineering flows.

What should I know about Bernoulli's Equation and Pitot Tubes?

A Pitot tube is a simple device, used on aircraft and in wind tunnels, that applies Bernoulli's equation directly to measure airspeed. It has two pressure ports: one facing directly into the airflow, which brings the air to a stop and measures the 'stagnation' or total pressure, and one mounted flush with the surface, which measures the undisturbed 'static' pressure of the moving air. Applying Bernoulli's equation between a point far upstream (moving at the flight speed v, with static pressure p) and the stagnation point (where velocity drops to zero) gives p₀ = p + ½ρv², so the difference between stagnation and static pressure, Δp = p₀ − p, equals ½ρv². Rearranging gives v = √(2Δp/ρ), where ρ is the air density. A pressure sensor connected to both ports therefore converts a pressure difference directly into an airspeed reading. This is exactly why aircraft airspeed indicators are also called Pitot-static instruments, and why a blocked or iced-over Pitot tube — a known cause of several historical accidents — gives dangerously incorrect airspeed readings even though the aircraft's actual Bernoulli flow is unaffected.

Worked Example: Venturi Meter Pressure Drop

Water (density ρ = 1000 kg/m³) flows through a pipe that narrows from a cross-sectional area A₁ = 0.02 m² to A₂ = 0.005 m². The upstream velocity is measured at v₁ = 1.0 m/s. First, find the velocity in the narrow section using the continuity equation A₁v₁ = A₂v₂: v₂ = v₁ × A₁/A₂ = 1.0 × (0.02/0.005) = 4.0 m/s The flow speeds up fourfold as the area shrinks to a quarter. Now apply Bernoulli's equation (assuming the pipe is horizontal, so height terms cancel) to find the pressure drop between the wide and narrow sections: Δp = p₁ - p₂ = ½ρ(v₂² - v₁²) = 0.5 × 1000 × (4.0² - 1.0²) = 0.5 × 1000 × (16 - 1) = 7,500 Pa So the pressure in the narrow throat is 7.5 kPa lower than in the wide section — this pressure drop is exactly what a Venturi flow meter measures to infer flow rate, and it's the same physical mechanism that lets a carburettor draw fuel into a fast-moving air stream.

Bernoulli's Equation: Terms and Typical Magnitudes

In most everyday plumbing and HVAC problems the dynamic pressure term is much smaller than the static pressure, which is why pressure drop across a Venturi constriction (a few kPa) looks small compared to mains water pressure (hundreds of kPa) — but it's still large enough to drive measurable flow-rate instruments, carburettors, and aspirators. In aerodynamics the ordering flips: dynamic pressure from fast-moving air is the dominant term used to generate lift, because air's low density is offset by much higher velocities (100+ m/s for aircraft versus 1-5 m/s for pipe water).

Worked Example: Pitot Tube Airspeed Calculation

An aircraft's pitot tube measures a dynamic pressure of 500 Pa in air of density ρ = 1.225 kg/m³ (sea-level standard conditions). Since dynamic pressure is ½ρv², solving for airspeed: v = √(2 × dynamic pressure / ρ) = √(2 × 500 / 1.225) = √(816.3) ≈ 28.6 m/s Converting to more familiar units: 28.6 m/s × 3.6 ≈ 102.9 km/h. This is exactly how a pitot-static system (the small forward-facing tube visible on aircraft and some vehicles) converts a pressure difference into an airspeed reading, using Bernoulli's equation in reverse — measuring pressure to infer velocity, rather than measuring velocity to infer pressure as in the earlier Venturi example. Because air density falls with altitude, the same dynamic pressure at high altitude corresponds to a higher true airspeed than at sea level, which is why aircraft instruments distinguish 'indicated airspeed' (what the pitot tube directly implies at sea-level density) from 'true airspeed' (the actual speed through the air at that altitude).

When Bernoulli's Equation Does Not Apply

Bernoulli's equation assumes the flow is steady, incompressible, non-viscous (no energy lost to friction), and follows a single streamline. Real fluids violate at least one of these assumptions to some degree — viscosity dissipates energy as heat, turbulence mixes streamlines together, and gases compress noticeably at high speed. For slow-moving liquids in short, smooth pipes, the idealised equation is usually accurate to within a few percent. But for long pipe runs with significant friction, or for gas flows approaching the speed of sound, engineers add correction terms (friction head loss, compressibility factors) or abandon Bernoulli's equation altogether in favour of more complete fluid dynamics models.