Bernoulli's Principle (Ideal Flow)

Fluid dynamics simulator for steady-state flow. Assumes an incompressible, inviscid fluid (no friction or roughness losses).

Pressure 1

Absolute pressure at the pipe inlet

01000

Velocity 1

Fluid velocity at the inlet

0100

Height 1

Height of the geometric center of section 1

0100

Diameter 1

Inside diameter of the pipe at point 1

0100

Results

Velocity 2

m/s

Δv: +6 m/s

Pressure 2

71.33

kPa

ΔP: -30 kPa

Flow Rate

0.01571

m³/s

Fluid Dynamics (Bernoulli's Equation)

Fluid

Thermodynamic properties (density and vapor pressure) assumed at 20 °C and 1 atm.

Fundamentals & Explanation

Bernoulli's equation: energy conservation in a fluid

Bernoulli's equation is the principle of conservation of energy applied to a moving fluid. Along a streamline, the total energy per unit volume splits into three terms that trade off against one another while their sum stays constant:

Pressure energy (P): the flow work exerted by the fluid on the adjacent volume.
Kinetic energy (½ρv²): proportional to the square of the velocity.
Potential energy (ρgh): tied to the height above a reference datum.

P_1 + \tfrac{1}{2}\rho\, v_1^2 + \rho\, g\, h_1 = P_2 + \tfrac{1}{2}\rho\, v_2^2 + \rho\, g\, h_2

Dividing by ρg recasts the three terms as heads (pressure, velocity and elevation head), the form used in hydraulic engineering. This simulator works in pressure units, but both readings are equivalent.

Continuity and the Venturi effect

The continuity equation expresses conservation of mass: the same flow rate Q = A·v must cross every section, so a smaller area forces a proportionally higher velocity (v ∝ 1/A ∝ 1/d²).

A_1 \cdot v_1 = A_2 \cdot v_2

Combined with Bernoulli, this velocity increase in a constriction forces the static pressure to drop: the Venturi effect. It can feel counter-intuitive —the fluid speeds up yet its pressure falls—, but it follows directly from the total energy being conserved.

Static, dynamic and stagnation pressure

Bernoulli's terms all carry units of pressure, and it helps to name them: the static pressure (P) is what a gauge moving with the fluid would read —what we report as P₂—; the dynamic pressure (½ρv²) is the part tied to motion. Their sum is the stagnation pressure, constant in horizontal flow:

P_0 = P + \tfrac{1}{2}\rho\, v^2 = \text{const}

When velocity rises, static pressure must fall. Bringing the fluid to rest recovers P₀: that is the principle of the Pitot tube, which measures velocity from P₀ − P = ½ρv².

Physical limiting cases

The pressure drop in a constriction cannot continue without bound; physics imposes two boundaries that the simulator flags:

Cavitation:if a liquid's static pressure falls to its vapor pressure, it boils locally, forming bubbles that collapse downstream and damage pumps and propellers. P₂ is compared with the vapor pressure at 20 °C.
Compressibility: in gases, once Ma = v/c > 0.3 density stops being constant and the incompressible form loses validity.
Unreachable vacuum: absolute pressure cannot be negative; a configuration demanding it is impossible —cavitation would set in first—, so it is blocked.

Applications

Wing lift:air moves faster over the upper surface, where pressure is lower, producing an upward force. (The higher speed comes from circulation and angle of attack, not from the “equal transit time” myth.)
Venturi meter and orifice plate: infer the flow rate from the pressure drop across a calibrated constriction.
Atomizers and carburetors: a fast air stream lowers the pressure and draws liquid up from a reservoir.
Torricelli's theorem: the efflux speed from an orifice, v = √(2gh), is the special case of Bernoulli between the free surface and the orifice.

Worked example

Water in a horizontal pipe narrowing from d₁ = 10 cm to d₂ = 5 cm, with P₁ = 101.3 kPa and v₁ = 2 m/s:

Continuity (A ∝ d²): halving the diameter cuts the area to ¼, so v₂ = 4 · 2 = 8 m/s.
Flow rate: Q = A₁v₁ = π(0.05)²·2 ≈ 0.0157 m³/s.
Bernoulli (h₁ = h₂): P₂ = 101 325 + ½·1000·(4 − 64) = 71.3 kPa.

The velocity quadruples and the pressure drops by 30 kPa: the Venturi effect in numbers.

Scope and limitations

The model assumes an incompressible fluid (constant ρ), in steady state and inviscid (friction losses neglected), with no external work from pumps or turbines. Long runs or viscous fluids need the energy equation with the Darcy–Weisbach term —the topic of a separate head-loss calculator.

The diagram is schematic: diameters and heights are normalized to keep the conceptual behavior readable on any screen, not drawn to a 1:1 scale.

Quantity	Symbol	SI Unit
Pressure	P	pascal (Pa)
Velocity	v	m/s
Height	h	metre (m)
Density	ρ	kg/m³
Flow rate	Q	m³/s

References

Çengel, Y. A., & Cimbala, J. M. Fluid Mechanics, Ch. 5.
Munson, Young & Okiishi. Fundamentals of Fluid Mechanics, Ch. 3.
White, F. M. Fluid Mechanics (McGraw-Hill).

Bernoulli's Principle — Wikipedia

Bernoulli's Principle (Ideal Flow)

Results

Fluid Dynamics (Bernoulli's Equation)

Fundamentals & Explanation

Bernoulli's equation: energy conservation in a fluid

Continuity and the Venturi effect

Static, dynamic and stagnation pressure

Physical limiting cases

Applications

Worked example

Scope and limitations

References

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Bernoulli's equation: energy conservation in a fluid

Continuity and the Venturi effect

Static, dynamic and stagnation pressure

Physical limiting cases

Applications

Worked example

Scope and limitations

References