Globe in Motion · Flow

Reynolds Number of Everything

From bacteria to hurricanes: why every flow looks the way it does

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Each step along the rule is ten times the last. Left of the break the scale is squeezed to fit the slowest flows on Earth. Choose any example, or use the calculator below to place your own flow, marked in red.

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Your own flow · Re

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Slide along a tenfold scale, or type a number.

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IOne number, two forces

Left, creeping flow round a ball: the streamlines close up behind it exactly as they opened in front. Right, a high Reynolds number: the flow peels away and leaves a churning wake.

Re = inertia ÷ viscous force = v L ÷ ν

Every moving fluid is a contest between two things. Inertia is its wish to keep going the way it is going; viscosity is the friction between neighbouring layers that smooths every motion out. The Reynolds number is the ratio of the two, and it needs only three ingredients: the speed v, a size L, and the fluid’s kinematic viscosity ν, its stickiness divided by its density.

When Re is small, viscosity wins: the flow is smooth, orderly and reversible, and stops the instant you stop pushing. When Re is large, inertia wins: little disturbances grow instead of dying out, and the flow breaks into eddies within eddies. Osborne Reynolds found the threshold in 1883 by running dye into water flowing through a glass pipe: below about 2,300 the streak stayed a thread; above it, it burst into a cloud.

Re has no units, so a bacterium, a cyclist and a hurricane can all be set on one scale.

IILife at low Reynolds number

A scallop that opens slowly and shuts fast gets nowhere at low Re: it ends each cycle where it began. A corkscrew turning one way, like a bacterium’s flagellum, does make progress.

A bacterium lives at a Reynolds number of about 10−5. Water is to it as treacle would be to us, only far more so: if it stopped swimming it would come to rest in less than the width of an atom. There is no gliding, no coasting, no momentum to borrow.

The physicist E. M. Purcell put a striking consequence in a lecture of 1977. In creeping flow only the shape of a stroke matters, not its speed, so any motion that simply retraces its path, such as a hinge opening and closing, carries a swimmer back exactly as far as it carried it forward. He called it the scallop theorem: a scallop could not swim at low Reynolds number, however quickly it clapped.

Micro-organisms escape it with strokes that are not reversible: the rotating corkscrew of a bacterium, the travelling wave down a sperm’s tail, the out-of-step beating of cilia.

Reference: E. M. Purcell, “Life at low Reynolds number”, American Journal of Physics 45, 3–11 (1977).

IIISame Re, same flow

Two flows with the same shape and the same Reynolds number look the same, whatever their size or fluid. This is dynamic similarity, and it is why a model in a wind tunnel can stand in for a full-sized aircraft, and why a slow drop of glycerol can mimic a fast one of water.

The catch is that a smaller model needs a faster flow, or a thinner fluid, to keep Re the same. The table shows what it would take. Real test tunnels cheat with cold, dense nitrogen under pressure, and ship tanks match the waves rather than Re, then correct for friction.

To watch shapes in a stream yourself, open the Flow Window.