Fluid Flow and Bernoulli's Principle Simulator
Equations in this simulation
| Q | Flow rate Q (liters per second) | the volume passing any section each second; with an incompressible fluid it is the same in every section | |
|---|---|---|---|
| A | cross-section of the pipe | from Pipe diameter D₁ (cm) and Throat diameter D₂ (cm) | |
| v | mean speed of the fluid in a section | halve the diameter and the area falls to a quarter, so the speed goes up four times |
With the current values:
| P | pressure in a section | what the column above it shows, in kPa above the atmosphere | |
|---|---|---|---|
| ρ | density of the fluid | set by the Fluid menu | |
| g | gravitational acceleration | 9.81 m/s² | |
| z | height of the section | from Height of the outlet above the inlet h (m) | |
| ΔP_loss | pressure lost to friction | zero unless Include friction (viscosity) is ticked |
With the current values:
| ΔP | pressure drop from the pipe to the throat | a Venturi meter turns this drop back into a flow rate; friction makes the reading slightly high |
|---|
With the current values:
| Re | Reynolds number | below about 2,300 the flow in a pipe is laminar, above about 4,000 turbulent | |
|---|---|---|---|
| μ | viscosity of the fluid | set by the Fluid menu |
With the current values:
| f | Darcy friction factor | f = 64 ÷ Re is the Hagen–Poiseuille law, ΔP = 8 μ L Q ÷ (π R⁴); the turbulent one is the Blasius formula for a smooth pipe |
|---|
With the current values:
How to use the Bernoulli's principle simulator
- Pick a fluid and set the flow rate, the pipe and throat diameters, the height of the outlet and the pressure at the inlet. The dots stream through the pipe; they speed up in the narrow throat, because the same volume has to pass every section each second.
- Read the three pressure columns over sections A, B and C, like the tubes of a Venturi meter. The pressure falls where the fluid speeds up and where the pipe rises, as Bernoulli's equation says; the columns stretch the differences between the gauges so small drops show, and each is labeled in kPa. The chart plots the pressure along the whole pipe, and the dashed line, pressure plus ½ρv² plus ρgz, stays flat while there is no friction.
- Tick Include friction to give the fluid viscosity: the flow takes a parabolic profile when it is laminar and a flatter one when it is turbulent, and the dashed line now slopes down. Try honey to see laminar flow with a large pressure loss, or narrow the throat until the pressure there drops below the vapor pressure and the model warns of cavitation. The Pipe Flow Calculator and the Reynolds Number Calculator do the same sums for real pipes.
Frequently asked questions
What does Bernoulli's principle say?
Along a streamline in a steady flow of an incompressible fluid without friction, P + ½ρv² + ρgz is constant: pressure, kinetic energy per volume and potential energy per volume add up to the same total. So where the fluid moves faster, or higher, its pressure is lower. It is conservation of energy for a moving fluid.
Why does the fluid speed up in the narrow part?
Because of continuity. An incompressible fluid cannot pile up anywhere, so the volume flow Q = A v is the same through every section. Halve the diameter and the area falls to a quarter, so the speed rises four times.
How does a Venturi meter measure flow?
It measures the pressure drop between the pipe and a narrower throat. With continuity and Bernoulli's equation, Q = A₂ √(2ΔP ÷ (ρ(1 − (A₂/A₁)²))). Real meters multiply this by a discharge coefficient of about 0.98 to allow for friction.
What is the Reynolds number?
Re = ρvD ÷ μ, the ratio of inertial to viscous forces in the flow. In a pipe, flow is usually laminar (smooth layers) below about 2,300 and turbulent (chaotic eddies) above about 4,000. Water in a household pipe is almost always turbulent; honey and blood in capillaries are laminar.
Is this how airplane wings make lift?
Partly. Air does move faster over the top of a wing and the pressure there is lower, which Bernoulli's equation describes. But the popular explanation that air over the top must "catch up" with air underneath is wrong; the speeds come from the wing turning the airflow downward, and explaining that needs the full equations of fluid flow, not Bernoulli's principle alone.
It says WebGL is turned off.
The 3D view needs WebGL, which every current browser has. It can be switched off by hardware acceleration being disabled in the browser settings, or by a very old graphics driver. Turn hardware acceleration on, or try another browser.
Is anything uploaded?
No. The simulation is drawn by your own browser with WebGL; nothing is sent anywhere, and it keeps working offline once the page has loaded.