This simulation recreates Osborne Reynolds' classic 1883 dye-injection experiment in pipe flow. Coloured tracer streams are released at the left wall and carried downstream by a velocity field that depends on the Reynolds number, Re = ρUL/μ = UL/ν. Below Re ≈ 2300 the field follows a smooth parabolic profile; above it the mean follows a 1/7th power law with added random fluctuations, producing the chaotic mixing characteristic of turbulence.
The Reynolds-number slider (200–12000) sets the flow regime and updates the live badge, turbulent intensity and profile readout. The stream-count slider (4–16) adds or removes coloured dye filaments, while the flow-speed slider scales the bulk advection. Watching streams stay parallel or break apart explains why pipelines, blood vessels, aircraft boundary layers and rivers behave so differently as speed, size or viscosity change.
What does this simulation actually show?
It shows coloured dye streams flowing through a channel, just like Reynolds' original experiment. At low Reynolds numbers the streams stay neat and parallel (laminar); as you raise the slider past the transition they wobble, spread and mix into one another (turbulent). A side panel draws the matching velocity profile across the channel.
What is the Reynolds number?
The Reynolds number is a dimensionless ratio of inertial to viscous forces, Re = ρUL/μ = UL/ν, where U is a characteristic speed, L a characteristic length, and ν the kinematic viscosity. It predicts the flow regime: small Re means viscosity dominates and damps disturbances, while large Re lets inertia amplify them into turbulence.
When does flow become turbulent in this model?
The simulation treats Re below 2300 as laminar, 2300 to 4000 as transitional, and above 4000 as fully turbulent. These are the textbook thresholds for flow in a circular pipe. The badge and statistics panel switch colour and label automatically as you cross each boundary.
The Reynolds-number slider (200 to 12000) selects the flow regime and drives the visual mixing. The stream-count slider (4 to 16) sets how many coloured dye filaments are injected. The flow-speed slider (0.3× to 3.0×) scales the overall advection speed of the particles without changing the regime.
For steady laminar flow the velocity varies as u = u_max(1 − r²/R²), a parabola that is zero at the walls and maximum at the centre. This is the Hagen–Poiseuille profile, and the simulation uses it directly to advect the tracer particles whenever Re is below 2300.
Above the transition the mean velocity is approximated by the empirical 1/7th power law, u ≈ u_max(1 − |r/R|)^(1/7). Turbulent mixing transports momentum efficiently across the channel, so the profile is much flatter in the core and steeper near the walls than the laminar parabola, giving a fuller, blunter shape.
It is a qualitative, illustrative model rather than a full solution of the Navier–Stokes equations. The mean follows the correct power-law shape, and random velocity fluctuations whose intensity grows with Re mimic turbulent unsteadiness, but it does not resolve true eddies or compute pressure. It is built for intuition, not engineering prediction.
It is a percentage that scales the random fluctuation added on top of the mean velocity once Re exceeds 2300. In the model it rises with Reynolds number and is capped at about 22%, representing the root-mean-square velocity fluctuation as a fraction of the bulk speed. At laminar Reynolds numbers it stays at 0%.
At low Reynolds numbers viscous forces are strong relative to inertia, so any small wobble in the dye is smoothed out before it can grow. As Re increases, inertia begins to dominate and tiny disturbances feed on the mean shear, amplifying until the orderly streams break down into chaotic three-dimensional motion.
It governs drag and mixing in pipelines, the onset of turbulent blood flow in arteries, heat exchanger performance, the boundary layer over aircraft wings, and the behaviour of rivers and pumps. Engineers often design to keep flow laminar for low drag, or deliberately trip it turbulent to enhance mixing and delay separation.
Between Re 2300 and 4000 the flow is transitional: it flickers intermittently between laminar and turbulent patches and is highly sensitive to disturbances, roughness and inlet conditions. Above Re 4000 the simulation labels it fully turbulent, with sustained chaotic mixing and the characteristic flatter 1/7th power-law mean profile.