Bernoulli's Equation Flow Visualizer
What's on screen
A converging-diverging pipe (Venturi shape) with animated speed-colored particles flowing through it: blue at the wide inlet (slow), shifting to red at the narrow throat (fast), back to blue at the wide outlet.
Above the pipe, manometer tubes rise from 5 tap points showing static pressure as water column height (shorter column = lower pressure at the throat). Below the pipe, stacked energy bars at 7 stations show how pressure energy (blue), kinetic energy (red), and potential energy (green) trade off while total energy (dashed line) stays constant.
Tap or drag anywhere on the pipe to move a golden probe that reads local P, v, and h values.
The pipe shape uses a smooth cubic interpolation (smoothstep) for the contraction and expansion, so there are no sharp edges that would cause separation in real flow.
4 sliders:
V₁ (0.2 to 10.0 m/s): Inlet velocity. Higher inlet = proportionally higher throat velocity. At V₁=5 and D₂/D₁=0.25, throat velocity hits 80 m/s and pressure can go negative (cavitation warning).
D₂/D₁ (0.15 to 0.95): Throat-to-inlet diameter ratio. The area ratio is the square: D₂/D₁=0.50 means A₂/A₁=0.25, so v₂=4×v₁. Smaller ratio = more dramatic pressure drop.
Δh (-3 to 3 m): Elevation change from inlet to outlet. Positive = uphill (pipe tilts, pressure drops even without area change). Negative = downhill (siphon behavior, pressure can recover).
P₁ (50 to 500 kPa): Inlet pressure. Sets the baseline. Higher P₁ means the throat can sustain higher velocities before pressure goes negative.
4 toggle overlays:
Manometers: Vertical tubes from pipe wall showing static pressure as liquid column height. The height difference between inlet and throat manometers is ΔP/(ρg), which is exactly what a real Venturi meter measures.
Particles: Speed-colored dots. Blue = slow (wide section), red = fast (throat). Particles bunch together at the inlet and spread apart at the throat because they move faster through the narrow section.
Energy bars: 7 stacked bars showing pressure/kinetic/potential breakdown. All bars have the same total height (constant total energy). As the pipe narrows, blue shrinks and red grows. As elevation increases, green grows and blue shrinks.
Velocity vectors: Arrow arrays inside the pipe showing local flow speed and direction. Arrows are longer at the throat and shorter at the inlet/outlet.
The probe (tap anywhere):
Drag your finger or mouse along the pipe. The golden probe line follows. The metrics bar at the top updates live with P, v, h, area ratio, and total head at that exact position. Move from inlet to throat: watch P drop from 200 to ~170 kPa while v jumps from 2.0 to 8.0 m/s. The total head H stays constant. This is Bernoulli in your fingertip.
Key slider experiments::
Default Venturi: V₁=2.0, D₂/D₁=0.50. Probe the inlet: P=200 kPa, v=2.0 m/s. Probe the throat: P=176 kPa, v=8.0 m/s. The 24 kPa pressure drop is what drives the manometer difference. A real Venturi flowmeter measures exactly this ΔP to calculate Q.
Switch to fire hose nozzle (D₂/D₁=0.25). Area ratio = 0.0625. Throat velocity = 24 m/s from a lazy 1.5 m/s inlet. Nearly all 400 kPa of inlet pressure converts to kinetic energy. The energy bar at the exit is almost entirely red. This is how firefighters get a 24 m/s jet from a low-pressure hose.
Set elevation Δh = +10 m. Even without any area change, pressure drops by ρgΔh = 98 kPa. The green (potential) energy bars grow from left to right while blue (pressure) bars shrink. This is why your shower pressure drops on the top floor.
Drag D₂/D₁ to 0.15 with V₁=5.0 and P₁=200. The throat velocity exceeds 200 m/s and throat pressure goes deeply negative. The info bar warns "cavitation risk." In reality, the fluid would vaporize, forming bubbles that collapse violently and damage the pipe walls. This is the practical limit of Bernoulli: you cannot reduce pressure below the vapor pressure.
Switch to siphon (Δh=-5m). Pressure at the highest point (throat) drops, but the downhill exit recovers it. The pipe effectively lifts fluid over a barrier using only gravity. No pump needed. If the high point pressure drops below atmospheric minus ρg×height, the siphon breaks.