Bernoulli & Venturi Calculator

Calculate pressure and velocity changes using Bernoulli's equation for ideal fluid flow.

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Bernoulli Connects Pressure, Speed, and Elevation

Bernoulli’s equation is a mechanical-energy balance for steady, incompressible, low-viscosity flow along a streamline. The terms p, ½ρv², and ρgh represent pressure energy, kinetic energy, and gravitational potential energy per unit volume. If elevation is unchanged, faster ideal flow is associated with lower static pressure when no pump, turbine, or major loss is present.

A Venturi meter combines Bernoulli with continuity. Narrowing the area raises speed, and the resulting pressure drop can be used to infer flow rate. Real pipes have friction and nonuniform velocity profiles, so discharge coefficients or head-loss terms are often required for accurate engineering measurements.

p+½ρv2+ρgh=constant,   A1v1=A2v2
SymbolMeaningWhy it appears / units
pStatic pressurePa; thermodynamic pressure of the moving fluid.
ρFluid densitykg/m³; assumed constant in incompressible form.
vFlow speedm/s; kinetic-energy term scales with v².
hElevationm; gravitational potential term.

A pressure decrease in a constriction is not caused by continuity alone. Continuity sets the speed change; Bernoulli relates that speed change to pressure and elevation under the ideal assumptions.

For a Venturi check, pair Bernoulli with continuity. If an incompressible pipe narrows, A2<A1 implies v2>v1. At equal elevation and with negligible losses, the faster section should have lower static pressure. A result showing both velocity and pressure increasing in that case signals a setup error.

Worked Examples

Example 1: Venturi: v₁=1m/s → v₂=5m/s, P₁=200kPa, water
P₂=200000+0.5×1000×(1-25)
Result: P₂=188kPa — pressure drops 12kPa
Venturi meter: pressure drop measures flow
Example 2: Airplane wing: faster air above (v₂=60m/s) vs below (v₁=50m/s), air
P₂=P₁+0.5×1.225×(2500-3600)
Result: ΔP=674 Pa lift per m²
Lift from Bernoulli pressure difference
Example 3: Horizontal speed increase
water, v1=2m/s, v2=5m/s
Result: p1−p2≈10.5kPa
The higher-speed section has lower static pressure in ideal horizontal flow.
Example 4: Area contraction
A2=0.5A1
Result: v2=2v1
Continuity turns a halved area into doubled average speed for incompressible steady flow.

Common Mistakes

⚠️
Applying Bernoulli across large friction losses without correction

Real viscous losses reduce mechanical energy and require a head-loss term.

⚠️
Confusing static pressure with stagnation pressure

Stagnation pressure includes the kinetic-energy contribution obtained by ideally slowing the flow to rest.

⚠️
Using the incompressible equation for high-speed gas flow

When density changes significantly, compressible-flow relations are required.

Frequently Asked Questions

Why does fast-moving fluid have lower pressure?
Conservation of energy: kinetic energy + pressure energy = constant. Faster flow (more KE) means less pressure energy. This is Bernoulli's principle — used in aircraft wings, Venturi meters, carburetors, and atomizers.
Bernoulli limitations?
Assumes: incompressible flow, inviscid (no viscosity), steady, along streamline. Fails near boundary layers, in turbulent regions, with heat transfer, or compressible flow. Add Darcy-Weisbach for pipe losses.
Does faster flow always mean lower pressure?
Not universally. That trend follows under specific Bernoulli conditions along a streamline with comparable elevation and no major energy addition or loss. Pumps, friction, geometry, and unsteady effects can change the relationship.
Why is a Venturi useful for flow measurement?
The constriction creates a predictable speed increase and pressure drop. Measuring the pressure difference and applying continuity plus Bernoulli allows flow rate to be estimated.
What is stagnation pressure?
It is the pressure a fluid would reach if brought to rest ideally and isentropically. For incompressible horizontal flow, p0=p+½ρv².
Can Bernoulli be used between two different streamlines?
Only under additional conditions, such as irrotational flow, where the Bernoulli constant is the same across streamlines. The safest introductory use is along one streamline.
When can the elevation term be omitted from Bernoulli’s equation?
The elevation term ρgz can be omitted between two points at the same height, or when the height difference is negligible compared with the pressure and velocity-head changes being studied. If the pipe rises or falls appreciably, include z; otherwise the predicted pressure change can have the wrong magnitude or even the wrong direction.

Formula Explorer connections

Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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