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Open a tap a little and the water falls as a smooth, glassy column. Open it further and the stream turns ragged. In 1883 Osborne Reynolds turned that everyday change into a precise result, using a glass tube, a tank of still water and a thin thread of dye. The number he found still decides how engineers design pipes. This post covers who he was, what he saw, how to use the Reynolds number, and what his approach teaches.
Who was Osborne Reynolds?
His MacTutor biography gives his dates as 23 August 1842, in Belfast, to 21 February 1912, in Watchet, Somerset. Before university he served an apprenticeship with an engineering firm, then read mathematics at Cambridge, graduating in 1867 as Seventh Wrangler.
In 1868 he became the first professor of engineering at Owens College, Manchester, which later became the University of Manchester. MacTutor describes it as only the second such chair in England. He was elected a Fellow of the Royal Society in 1877 and later received its Royal Medal. Besides turbulence, he developed a theory of lubrication in 1886 and studied heat transfer, turbines and the regime of rivers.
What did the dye experiment show?
The paper is An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, in the Philosophical Transactions for 1883. By "direct" he meant smooth flow parallel to the pipe; by "sinuous" he meant flow crossing and recrossing it. The text is reprinted in his collected papers, volume 2.
His final apparatus used a glass tube nearly 5 feet long and about an inch wide, chosen from many for its uniform bore, with a varnished wooden trumpet at the inlet. It lay inside a tank of water that had stood for hours to settle. A fine tube fed coloured water into the trumpet, and a valve far below controlled the flow.
At low speeds the colour band stayed "beautifully steady". Then, on a slight further opening of the valve, it suddenly spread and filled the tube with a coloured cloud. Lit by the flash of a spark, the cloud resolved into distinct eddies. Reynolds wrote that "it was a matter of surprise to me to see the sudden force with which the eddies sprang into existence."
He also saw that:
the disturbance never began nearer the inlet than about 30 tube diameters;
in smaller tubes it came and went in "flashes" before becoming general;
the critical velocity was proportional to the viscosity and inversely proportional to the diameter of the tube;
a coil of wire placed in the tube made eddies appear at less than half the usual critical velocity.
In a 1895 paper, reprinted in the same volume, he stated the criterion directly: the combination of diameter, mean speed, density and viscosity has, for round pipes, a critical value between 1,900 and 2,000, matching his own experiments and older ones by Poiseuille and Darcy. That combination is now called the Reynolds number.
How do you use the Reynolds number?
Streamline and turbulent flow and critical velocity appear in the JEE (Main) 2026 syllabus. The NCERT Class 11 fluids chapter in its current reprint describes a critical speed beyond which steady flow becomes turbulent, but does not name the Reynolds number, so learn it as the idea behind that critical speed. NCERT's table gives the viscosity of water at 20 °C as 1.0 mPa s, or 1.0 × 10⁻³ Pa s.
Worked example 1. Water at 20 °C flows at 0.50 m/s through a pipe 1.5 cm across. Re = ρvD/η = 1000 × 0.50 × 0.015 / (1.0 × 10⁻³) = 7,500. That is far above 2,000, so expect turbulent flow.
Worked example 2. What is the highest speed for reliably smooth flow in a 1.0 cm pipe? Set Re = 2,000: v = Re × η / (ρD) = 2,000 × 1.0 × 10⁻³ / (1000 × 0.010) = 0.20 m/s.
Worked example 3, from Reynolds's notebook. In his first run with the final apparatus, on 22 February 1880, eddies appeared at about 0.627 m/s in the one-inch (0.0254 m) tube. Using the 20 °C viscosity gives Re ≈ 1000 × 0.627 × 0.0254 / (1.0 × 10⁻³) ≈ 16,000. His water was only about 8 °C (46 to 47 °F), so its viscosity was higher and the true value lower, but still well above 2,000. Very still water can stay smooth far past the usual limit, which is exactly why he found the critical value depended on how undisturbed the water was.
Common mistakes:
Using the radius instead of the diameter. The pipe criterion uses D. Using r halves Re.
Mixing viscosity units. Convert mPa s to Pa s before you calculate.
Treating 2,000 as a sharp switch. Below it, disturbances die away. Above it, smooth flow can survive for a while if nothing disturbs it, as example 3 shows.
The slow-flow end of this story, where viscosity dominates completely, belongs to George Stokes and the drag on a falling sphere.
What can a student learn from Reynolds's way of working?
Make the invisible visible. A thread of dye turned an argument about water into something you could watch.
Control your conditions. He chose the straightest tube and let the tank settle for hours. Careless set-up hides real effects.
Look for the combination, not the single cause. Speed alone did not decide the flow; the grouping ρvD/η did. Many JEE problems hinge on such combinations.
Report the surprise. He had expected eddies to appear gently and said plainly that they did not.
If you would rather see streamlines than imagine them, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
Next time you open a tap, watch for the change from smooth to ragged, then estimate the Reynolds number at that moment from the stream's width and speed.
