Laminar-to-turbulent transition, Kolmogorov theory, and the Millennium Prize Problem
Turbulence is everywhere: in the smoke rising from a candle, the wake behind a ship, the jet stream that shapes our weather, and the blood flowing through our arteries. It is the most common state of fluid motion in nature, yet it remains one of the deepest unsolved problems in physics and mathematics.
The transition from smooth, orderly laminar flow to chaotic turbulent flow is governed by the Reynolds number Re = UL/ν, the ratio of inertial forces to viscous forces. When Re is small, viscosity dominates and the flow is laminar. When Re is large, inertia dominates and the flow becomes turbulent.
“Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls, and so on to viscosity.”
— Lewis Fry Richardson (1922), paraphrasing Jonathan Swift
Osborne Reynolds's famous 1883 experiment injected a thin streak of dye into pipe flow and observed how it behaved as the flow rate increased. At low flow rates (Re < 2300), the dye remained a perfectly straight line — laminar flow with a parabolic velocity profile. At higher flow rates, the dye began to wobble and eventually broke apart into chaotic mixing.
Drag the Reynolds number slider to recreate Reynolds's experiment. Watch the cyan dye streak transition from a smooth laminar line to intermittent wobbles and finally to fully turbulent dispersion.
The cyan dye streak is injected at the pipe center. At low Re the parabolic velocity profile keeps the streak straight. Above Re ≈ 2300 intermittent bursts appear. Above Re ≈ 4000 the flow is fully turbulent and the dye disperses across the pipe.
Richardson's poem captures the central idea of turbulence: energy is injected at large scales (by stirring, wind, or pressure differences), then cascades through a hierarchy of ever-smaller eddies, until it finally reaches scales so tiny that molecular viscosity can convert the kinetic energy into heat. This is the energy cascade.
In 1941, Andrei Kolmogorov formalized this picture with his celebrated theory. In the inertial range — the vast span of scales between energy injection and viscous dissipation — the energy spectrum follows a universal power law: E(k) ∝ ε²⁄³ k¹⁻²⁄³, where ε is the energy dissipation rate and k is the wavenumber. The smallest scales are set by the Kolmogorov microscale η = (ν³/ε)¼.
Adjust the energy injection rate and viscosity to see how they shape the energy spectrum. Lowering viscosity extends the inertial range to smaller scales, increasing the Reynolds number of the flow.
Lower viscosity pushes the dissipation scale to smaller sizes, widening the inertial range where the k¹⁻²⁄³ power law holds. Higher energy injection raises the entire spectrum.
The Navier-Stokes equations have been used successfully for over 150 years to model fluid flow. Yet we still do not know, in three dimensions, whether smooth initial conditions always lead to smooth solutions — or whether the equations can develop a singularity (infinite velocity or infinite vorticity) in finite time.
In 2D the question is settled: Olga Ladyzhenskaya proved in 1969 that smooth solutions exist for all time. The key difference is that in 2D there is no vortex stretching — the mechanism by which vortex tubes in 3D can intensify without bound. In 3D, vortex stretching is present, and nobody has been able to prove that it cannot produce a blow-up.
Step through the explanation below and compare the 2D (smooth forever) and 3D (potential blow-up) scenarios side by side.
Start with an infinitely smooth (no sharp corners) initial velocity field v₀ that satisfies the incompressibility condition ∇·v = 0.
In the year 2000, the Clay Mathematics Institute designated seven Millennium Prize Problems, each carrying a reward of $1,000,000 for a correct solution. The Navier-Stokes existence and smoothness problem is one of them:
“Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes equations.”
As of today, only one of the seven Millennium Problems has been solved (the Poincaré Conjecture, by Grigori Perelman in 2003). The Navier-Stokes problem remains wide open — a testament to how much we still do not understand about the equations that govern every fluid flow around us.