One number decides which physics applies
A bacterium swimming in water and a whale swimming in water are not the same problem at different scales. They are different problems, and the thing that separates them is a single number with no units.
What the number is
The Reynolds number compares two effects that both act on moving fluid.
Inertia is the fluid’s reluctance to change what it is doing — its tendency to keep going, to overshoot, to carry momentum from one place to another.
Viscosity is the fluid’s internal friction — its tendency to smear velocity differences out, to bring neighbouring fluid to a common speed.
Density times speed times a length, divided by viscosity. The units cancel exactly, which is the whole point: what is left is a pure number that says which of the two effects is running the flow.
Why a ratio and not a size
The instinct is to think of small things and large things, or slow flows and fast ones. Neither is the right category, and the number says why.
A bacterium two micrometres long swimming at thirty micrometres a second has a Reynolds number around . A whale thirty metres long at ten metres a second has one around . Twelve orders of magnitude apart, and neither of them is characterised by its size alone: shrink the whale and slow it down enough and it would be in the bacterium’s world.
What matters is the combination. A large slow object in a viscous fluid and a small fast one in a thin fluid can share a regime exactly, and if they do, their flows are the same flow — geometrically scaled versions of one solution.
That is dynamic similarity, and it is the reason wind tunnels work at all. A model at one-tenth scale in a flow ten times faster has the same Reynolds number and therefore the same flow pattern, the same separation points, the same drag coefficient.
What changes at the thresholds
Crossing a threshold does not make the flow more of what it was. It makes it something else.
Below about one — creeping flow. Viscosity dominates completely. There is no wake, no separation, and no coasting: stop pushing and the motion stops instantly. Flow at this scale is also reversible, which has the strange consequence that a swimmer whose stroke looks the same forwards and backwards goes nowhere at all. Bacteria have corkscrew flagella rather than oars for exactly this reason.
Around ten to fifty — the first separation. A recirculating region appears behind a bluff body. Above roughly 47 for a cylinder, that wake becomes unsteady and starts shedding vortices alternately from each side.
Thousands to millions — the boundary layer regime. Viscosity has retreated into a thin film next to the surface and the rest of the flow is very nearly inviscid. This is where nearly all engineering lives.
Higher still — the layer itself becomes turbulent, which changes separation behaviour and can make drag fall as speed rises.
Life at the bottom of the scale
The creeping-flow world deserves more than a label, because almost nothing that holds in everyday experience holds there.
There is no coasting. Inertia is negligible, so the moment a bacterium stops beating its flagellum it stops moving — within about a tenth of a micrometre. Nothing glides. A bacterium that wants to travel must push continuously.
The flow is reversible. Run the driving motion backwards and the fluid retraces its path exactly. This has a startling consequence, usually called the scallop theorem: any swimmer whose stroke is the same forwards as backwards returns to where it started. A scallop, which opens slowly and snaps shut, works beautifully in water at human scale and would be motionless at a bacterium’s.
Mixing is impossible without stirring in a particular way. Two fluids brought together at low Reynolds number do not mix; they can be sheared apart and then, by reversing the shear exactly, brought back together with the interface restored. There is a well-known demonstration of this with dyed glycerine that looks like a conjuring trick and is simply the reversibility above.
None of this is exotic physics. It is the same Navier–Stokes equations with one term dominating instead of the other.
Every regime has its own number
Reynolds is the most important, and it is not alone. Each dimensionless group compares inertia with something else that could resist it, and each owns a threshold.
Mach compares the flow speed with the speed of sound, and decides whether the fluid has time to get out of the way — which is when density stops being a constant.
Froude compares inertia with gravity. Below one, a disturbance can send waves upstream; above one it cannot, and the flow is unaware of what is in front of it. That is the difference between a placid river and the sheet of water below a weir, and the abrupt jump between them is a hydraulic jump.
Strouhal compares a shedding frequency with the flow’s own timescale, and is why a wire hums in the wind at a pitch that rises with wind speed.
Weber compares inertia with surface tension, and decides whether a jet breaks into drops.
The pattern is the same every time: form a ratio, find that the units cancel, and discover that the number rather than any of its ingredients is what the physics responds to.
The drag crisis
One consequence is worth singling out because it is so counter-intuitive.
Over a narrow band of Reynolds number around , the drag coefficient of a sphere drops — by a factor of about four — as the flow gets faster.
Nothing about the sphere changed. What changed is that the boundary layer became turbulent before it reached the separation point. A turbulent layer carries more momentum near the wall and separates later, so the wake narrows abruptly and the pressure drag falls off a cliff.
This is the effect golf ball dimples exploit: they trip the layer early and pull the crisis down to the Reynolds numbers a struck ball actually flies at.
Why the units cancelling matters
The cancellation of units is not a tidy detail. It is the reason the number has any authority.
A quantity with units is a comparison against an arbitrary standard. Saying a flow is “fast” invites the question: fast compared with what? A metre per second is quick for treacle and imperceptible for air. Any statement about a flow that depends on the choice of metre or second cannot be a statement about the physics.
A dimensionless group has no such dependence. It compares the flow against itself — one term of its own governing equation against another — and the answer means the same thing to anybody, in any units, at any scale.
That is why the thresholds on the axis above are numbers rather than speeds. Shedding begins near Reynolds 47 for a cylinder whether the cylinder is a wire in a breeze or a chimney in a gale, and whether the measurements are in metres or feet.
The habit generalises far beyond fluids, and it is the single most transferable idea in the subject: when a problem seems to depend on too many quantities, look for the ratios in which the units cancel, and expect the physics to depend only on those.
Reading the axis honestly
A caution about the figure itself, because a regime axis is a compression and compressions mislead.
The bands have edges drawn on them and the physics does not. Separation does not begin at a sharp value; it begins over a range, and the range depends on the body’s shape, on the free stream’s turbulence and on the surface finish. A number quoted to two figures for a transition is a convenient fiction.
The axis is also specific to a circular cylinder. Quote the same thresholds for an aerofoil, a sphere or a flat plate and they move — sometimes by an order of magnitude.
And the length in the Reynolds number is a choice. Diameter is conventional for a cylinder, chord for a wing, hydraulic diameter for a duct, and a Reynolds number quoted without saying which is not a number at all. Comparing two values that used different conventions is the most common way to misuse the whole idea.
What the solver computed
The regime boundaries on the axis are quoted from the literature, and the site says so; they are experimental facts, not results of this solver.
What is computed here is the behaviour either side of one of them. lib/flow.js solves the flow past
a cylinder at a stated Reynolds number and the site measures whether reverse flow exists behind the
body. At Reynolds number 1 and 10 there is none. At 40 and 100 there is, and the recirculating region
grows. That transition is found rather than assumed, and it is the one threshold on the axis this
site can demonstrate directly.
The others are stated on the authority of measurements this site has not made, which is a distinction worth keeping visible.
Similarity has limits
Dynamic similarity is powerful and it is not free, and testing engineers spend a great deal of effort on where it breaks.
Matching the Reynolds number is only sufficient when it is the only relevant group. If compressibility matters, the Mach number has to match as well. If there is a free surface, the Froude number does. If there is surface tension, the Weber number does.
Matching two at once is usually impossible in a single facility. A ship model at the right Froude number for its waves is at quite the wrong Reynolds number for its skin friction, which is why ship testing splits the drag into two parts, scales each by a different rule, and adds them back together.
So the number is the organising idea of the subject and it is not the whole of it.
Wind tunnels, and the reason they work
Dynamic similarity is not a theoretical nicety; it is the entire justification for testing a model of something instead of the thing itself.
If a one-tenth-scale model is tested at the same Reynolds number as the full-size article, the two flows are the same flow. Not similar in spirit — identical, after scaling every length by ten and every velocity by the appropriate factor. Separation happens at the same place on the model as on the aircraft. The drag coefficient is the same number. The pressure coefficient at corresponding points is the same number.
Achieving it is the difficulty. To keep the Reynolds number up with a tenth-scale model, the tunnel must run ten times faster — which for an airliner model means going supersonic, and then the Mach number is wrong instead. The usual escapes are to raise the density by pressurising the tunnel, or to lower the viscosity by chilling the gas, both of which are expensive and both of which are done.
The alternative is to test at the wrong Reynolds number and correct afterwards, which works when the flow is not near a threshold and fails badly when it is. A model tested just below the drag crisis and an aircraft flying just above it are not describing the same physics at all.
The number in the figures on this site
A concrete illustration, since every viscous figure here carries a Reynolds number in its corner.
At Reynolds 1 and 10 the computed flow past a cylinder has no reverse flow anywhere: the wake closes, and the picture looks much like the ideal solution despite being solved with viscosity included. That is not a coincidence — at low Reynolds number the flow is nearly fore-and-aft symmetric for a different reason than the ideal theory’s, and the resemblance is misleading.
At Reynolds 40 and 100 there is reverse flow, the wake is open, and the pressure never recovers. The two pictures were produced by the same solver with one number changed.
Nothing about the cylinder changed between those two figures. The ratio did, and the physics followed it.
Where the model stops
The length in the formula is a choice. Diameter for a cylinder, chord for a wing, hydraulic diameter for a duct. Two Reynolds numbers are comparable only if they use the same convention, and quoted thresholds are meaningless without it.
The thresholds are approximate and geometry-dependent. Shedding starts near 47 for a circular cylinder and somewhere else for anything else.
Transition is not a number. Whether a boundary layer is turbulent depends on roughness, noise and pressure history, not on Reynolds number alone.
This site’s solver spans a narrow range. It resolves the attached-to-separated transition and nothing above it.
Who found it, and when
George Stokes had the group in 1851. Osborne Reynolds gave it its experimental meaning in 1883, in an apparatus that is still worth describing: a glass tube, water flowing through it, and a fine filament of dye introduced on the axis. At low flow the dye drew a straight line the length of the tube. Turn the flow up and at a definite point the filament broke into eddies and mixed.
Reynolds found that the transition happened at the same value of his group regardless of the tube’s diameter or the fluid’s viscosity — which is what established that the group, and not any of its ingredients, was the thing that mattered.
What the number cannot settle
Three limits, and they are the reason a Reynolds number is a starting point rather than an answer.
It does not give a force. Knowing a body is at Reynolds says which regime it is in and nothing about its drag; that still needs the flow to be solved or measured. The number classifies, it does not compute.
It does not know about shape. A cylinder and an aerofoil at the same Reynolds number are in different situations entirely, since where a flow separates depends on the pressure gradient and that depends on the body.
It does not survive on its own. Where compressibility, gravity or surface tension matter, matching Reynolds is necessary and not sufficient — and matching two groups at once is usually impossible in one facility.
So the honest description is that the Reynolds number tells a reader which physics to expect, and which of the thin-layer picture or the creeping-flow picture to reach for. That is a great deal, and it is not the answer.
The ladder from here
Next rungs: the Reynolds number itself in more detail, with the derivation and the choice of length scale; life at low Reynolds number and the scallop theorem; the drag crisis worked through; and dynamic similarity as the basis of model testing.
Then across to the other numbers — Mach, Froude, Strouhal, Weber — each of which owns a different threshold.