How this instrument works
Daniel Bernoulli published Hydrodynamica in 1738, arguing from vis viva — living force — in moving water. Leonhard Euler recast that argument around 1752 into pressure form, which is what engineers write today. Each term carries joules per cubic metre, units identical to pascals: static pressure P, dynamic pressure ½ρv², and elevation term ρgh. Along one streamline in steady flow that sum holds constant, so speeding flow up forces something else to yield — usually its static term.
Constancy of that sum is why pinching hose nozzles throws water further, and why Venturi throats pull fuel into carburettors. Instruments read it directly: a Pitot-static tube on an airliner senses dynamic head and reports airspeed; an orifice plate in process piping reads flow rate from measured drop. Magnitudes are worth memorising. Water at 2 m/s carries just 2 kPa of dynamic head, while one metre of elevation is worth 9.8 kPa, so geometry usually beats velocity in liquids. Air reverses that ranking — at 100 km/h its dynamic term reaches roughly 470 Pa, dwarfing any height contribution across a wing.
Four assumptions hold this equation together: steady flow, constant density, negligible viscosity, and two points sharing one streamline. Break any and answers drift. Viscous shear in a long pipe bleeds energy away even where speed and height never change, which is why plumbers size lines with Darcy-Weisbach instead. Past roughly Mach 0.3 a gas compresses enough to demand a different form. Across a pump, fan or turbine, work enters from outside and no streamline expression spans it.
- Enter Pressure at point 1 as absolute or gauge — either works, provided Pressure at point 2 gets read on that same datum.
- Set Speed at point 1 and Speed at point 2. Continuity supplies v₂ = v₁(d₁/d₂)², so halving pipe diameter quadruples speed.
- Give Height at point 1 and Height at point 2 against any reference level; only their difference enters this calculation.
- Enter Fluid density — 998 kg/m3 for water at 20 °C, 1025 for seawater, 1.225 for sea-level air.
- Read Pressure at point 2, switching its unit selector to kPa, bar, atm or psi as your gauges require.
Worked example — a 2 bar main through a Venturi throat
A 100 mm water main runs at 2 m/s and 200000 Pa absolute, or 2 bar. It necks into a 50 mm throat, one quarter of that area, so continuity forces flow to 8 m/s. Both taps sit level, h₁ = h₂ = 0, and density is 1000 kg/m3.
Substitution gives P₂ = 200000 + 0.5 × 1000 × (2² − 8²) = 200000 + 500 × (4 − 64) = 200000 − 30000 = 170000 Pa. So Pressure at point 2 reads 170 kPa, or 1.7 bar. Thirty kilopascals purchased sixteen times more kinetic energy density; where a diffuser downstream slows flow back to 2 m/s, most of that 30 kPa comes home again.
Questions
Why does pressure fall when a fluid speeds up?
Because total energy per unit volume stays fixed along a streamline, so any rise in kinetic share must be funded from somewhere, and a static term pays. Look at forces instead of energy and it reads plainer still: a parcel accelerating into a narrow throat is pushed from behind by higher pressure and meets something lower ahead. That imbalance is exactly what accelerates it. Cause and effect run both ways, and neither direction is mysterious.
Should I enter absolute or gauge pressure?
Either, so long as both entries share one convention. Only a difference appears in this equation, so a common 101325 Pa offset cancels exactly. Absolute values do matter when you check against vapour pressure for cavitation, or when gas density itself responds to loading. Mixing conventions — gauge at point 1, absolute at point 2 — throws answers off by precisely one atmosphere, an error that is easy to make and easy to miss.
When does this equation stop working?
Whenever friction, unsteadiness or compressibility grows significant. Real pipes shed energy to viscous shear even at constant diameter and level, and that loss belongs in Darcy-Weisbach rather than here. Gas flows past Mach 0.3 need a compressible form. Pumps, fans and turbines add or remove work, so no streamline expression spans them. Two points in separate regions — inside a wake and outside it, say — are joined by no streamline at all, which quietly invalidates most careless applications.
Does Bernoulli explain how a wing lifts?
Partly, though a popular version of that story is wrong. Air crossing a curved upper surface does move faster, static pressure there does drop, and integrating that drop over wing area yields lift. What fails is equal transit time: parcels splitting at a leading edge are said to rejoin at a trailing edge, which they do not — upper-surface air arrives sooner. Circulation and downwash give a fuller account, and NASA Glenn keeps a page devoted to that misconception.
How do I get Speed at point 2 from pipe sizes alone?
Use continuity. For constant density, A₁v₁ = A₂v₂, so v₂ = v₁(d₁/d₂)² in round pipe. Diameter enters squared, which is why halving it quadruples speed and raises a dynamic term sixteenfold. A 100 mm line at 2 m/s therefore reaches 8 m/s inside a 50 mm throat. Rectangular ducts use an area ratio directly rather than any diameter ratio.
What if Pressure at point 2 comes out near zero or negative?
Read that as a cavitation warning, not a physical prediction. No liquid holds absolute pressure below its own vapour point — near 2.3 kPa for water at 20 °C — and instead flashes into bubbles that collapse downstream with enough violence to pit impellers and valve seats. A negative absolute answer means your assumed speed change or elevation change is unreachable; real hardware chokes, cavitates or stalls first.