The Reynolds Number: From a Streak of Dye in 1883 to the Dimples on a Golf Ball

Open a kitchen tap just a little and the water falls as a clear, glassy rod. Open it further and at some point the stream turns ragged and starts to hiss. Same water, same tap, different kind of flow. The person who reduced “when does this happen?” to a single number was Osborne Reynolds, an engineering professor in Manchester.

The glass-tube experiment of 1883

In his 1883 paper in the Philosophical Transactions of the Royal Society, Reynolds laid long glass tubes in a tank of water and shaped their entrances into smooth bell mouths so the water would enter without a jolt. A thin nozzle in front of the entrance released a single thread of coloured water. A valve at the far end set the flow speed, and he watched what happened to the thread.

At low speed the dye stayed a straight, sharp line along the whole tube. As he opened the valve, the thread at some point began to waver and then burst across the whole cross-section, tinting all the water. Reynolds called the first kind of motion “direct” and the second “sinuous”. Today we say laminar and turbulent.

His key finding was that speed alone did not decide the switch. A wider tube moved the transition, and so did colder, more viscous water. Across tube diameters and water temperatures, the change always happened at the same value of one combination:

Re = ρ·U·D / μ = U·D / ν

Here ρ is density, U the mean velocity, D the tube diameter, μ the dynamic viscosity and ν = μ/ρ the kinematic viscosity. All the units cancel, so the number means the same thing for water or oil, a capillary or a water main. The name “Reynolds number” was attached to this combination later.

There is no single critical value

Textbooks often say pipe flow is laminar below Re = 2,300. Reynolds’s own results were more subtle. When he let the water in the tank settle for a long time and kept disturbances to a minimum, the flow stayed laminar up to about Re = 13,000. Going the other way, even when he disturbed the entrance deliberately, the flow calmed back to laminar downstream as long as Re was below roughly 2,000.

So about 2,000 is a floor: below it, turbulence cannot sustain itself in a pipe. Above it, when turbulence actually appears depends on the entrance shape, wall roughness and how large the incoming disturbances are. The figure of 2,300 is an engineering convention that allows for this. For anything other than a pipe, the value also depends on which length you call L, so a critical Reynolds number only means something when it states “for what, based on which length”.

The other half of the title: the law of resistance

The long title of Reynolds’s paper ends with “the law of resistance in parallel channels”, because he measured the pressure drop along the tube as well as watching the dye. In laminar flow the pressure difference needed to drive the water is directly proportional to the speed: double the flow and you double the loss, just as Hagen and Poiseuille had found in narrow tubes. Once the flow is turbulent the loss grows roughly with the 1.7th to 2nd power of speed, so doubling the flow through the same pipe needs something like three to four times the pressure. Eddies carry momentum toward the wall and steepen the velocity gradient there. That is why anyone designing water mains, pipelines or thinking about blood flow first asks which regime the flow is in.

A ratio of inertia to viscosity

One useful reading of Re is as a ratio between two tendencies: the fluid’s inertia, which keeps moving parcels moving, and viscosity, which makes neighbouring layers drag on each other and erase differences in speed. At low Re, any small wobble is smothered by viscosity almost at once. At high Re, wobbles grow faster than viscosity can damp them, and eddies spawn smaller eddies.

A few quick calculations show how wide the range is. They use ν ≈ 1.0×10⁻⁶ m²/s for water and 1.5×10⁻⁵ m²/s for air.

Situation	Speed	Length	Re
Bacterium swimming in water	30 μm/s	1 μm	about 3×10⁻⁵
Slow water in a 2 cm tube	0.1 m/s	2 cm	about 2,000
Household pipe, 15 mm	1 m/s	15 mm	about 15,000
Golf ball in flight	70 m/s	42.7 mm	about 2×10⁵
Car on a motorway	30 m/s	4.5 m	about 9×10⁶

To a bacterium, water is far thicker than honey is to us: the moment it stops swimming it stops moving. Water in household plumbing is mostly turbulent, and the air around a moving car certainly is.

Same shape does not mean same flow

A wind tunnel test tries to learn about a large object from a small model. Scaling the shape down exactly (geometric similarity) is not enough. For the flow pattern to match, the Reynolds number has to match as well (dynamic similarity). If it does not, a boundary layer that stays laminar on the model may be turbulent on the real thing, or separate at a different place, and the drag coefficient comes out wrong.

The difficulty is meeting that condition. Test a 1/10 model of a 4.5 m car travelling at 30 m/s in ordinary air and you need 300 m/s to match Re. That is close to the speed of sound, so now compressibility changes the flow instead. You cannot match Reynolds number and Mach number at the same time this way.

The way out is to change the fluid rather than the speed. Lower the kinematic viscosity ν and the same speed gives a larger Re.

• Compress the air: density rises and ν falls. That is the idea behind pressurised wind tunnels.
• Cool the gas: viscosity drops and density rises. NASA Langley’s National Transonic Facility injects liquid nitrogen and runs on nitrogen gas as cold as about −250 °F (around −157 °C), and NASA quotes Reynolds numbers up to about 44 million per metre.
• Use water: its ν is about 1/15 of air’s, so the same Re comes at 1/15 of the speed. Slow water channels are ideal for watching vortices.

Why golf balls have dimples

A smooth sphere’s drag coefficient sits around 0.4–0.5 for Reynolds numbers from the thousands to a few hundred thousand. Then, near Re ≈ 3×10⁵, it drops abruptly. This is the drag crisis.

While the thin boundary layer on the front of the sphere is laminar, it carries little energy. It cannot push against the rising pressure on the rear half and separates early, near the sides. A wide, low-pressure wake is left behind, and the pressure difference between front and back makes up most of the drag. Once the boundary layer is turbulent, faster fluid from further out is mixed down toward the wall. The layer clings longer, separates further back and leaves a narrower wake, so drag falls. Turbulence adds a little skin friction but removes much more pressure drag.

A golf ball off a driver moves at about 70 m/s, a Reynolds number around 2×10⁵. A smooth ball would still be just below the drag crisis, on the high-drag side. Dimples disturb the surface so that the boundary layer turns turbulent at a lower Re. Jin Choi, Woo-Pyung Jeon and Haecheon Choi at Seoul National University (2006) measured this in a wind tunnel with a 150 mm sphere. The flow separates briefly inside each dimple and reattaches, which boosts turbulence; the main separation is pushed back, and drag falls by as much as about 50% compared with a smooth sphere. They also showed why the lowered drag coefficient stays nearly constant across a range of Reynolds numbers, which is one reason a golfer’s distance does not swing wildly with small changes in ball speed.

Three common misconceptions

“Turbulence always means more drag.” In pipes it is broadly true: for the same flow rate, turbulent flow loses more pressure. For blunt bodies like spheres and cylinders, as above, a turbulent boundary layer can cut the total drag.

“Above 2,300 the flow is turbulent.” Reynolds himself kept laminar flow close to 13,000. The critical value is a range that moves with disturbances and entrance conditions.

“The Reynolds number is a property of the fluid.” It is a property of a flow: fluid, speed and length together. In the same air, the Reynolds number around a mosquito’s wing and around an airliner’s wing differ by a factor of roughly a million.

Seeing it in a simulation

In the Kármán vortex wind tunnel, set the wind very low and two twin vortices simply sit behind the cylinder. Raise it slowly and, somewhere around Re 50, they start to peel off alternately. That is the first transition in a cylinder wake described in Williamson’s 1996 review. The tool can only reach Re in the hundreds, but it shows the Reynolds and Strouhal numbers together, which makes the before-and-after easy to compare.

The 3D wind tunnel shows the real Re next to the Re the grid can actually carry. At the defaults the real value is in the millions and the grid’s is about a thousand. A sharp-edged flat plate still lands close to its textbook drag, but a smooth sphere comes out well above 0.47, with no sign of the drag crisis. It is the scale-model problem, reproduced inside a browser.

In smoke fluid, the viscosity slider works as a Reynolds-number dial turned backwards. Raise viscosity and the same stroke smears out like honey; drop it to zero and thin bands wind up into spirals.