Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Regimes and numbers

The Reynolds number, and the length in it

The most useful number in fluid mechanics has an arbitrary quantity buried in it, and quoting one without saying which length was used makes it meaningless. That detail is where most misuse comes from.
15 min read 6 figures One number decides the regimeWhat is conservedA smooth picture proves nothing

The Reynolds number is usually introduced as a definition to be memorised. It is better met as something that falls out of the equations when they are asked what they depend on.

Reynolds number: one number, four different flowsReynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.creepingattachedseparated, sheddingturbulentbacterium swimmingshedding begins, Re ≈ 47a thrown ballan airliner winga whaleReynolds numberinertia ÷ viscositylog₁₀ Rethe ratio decides the regime, not the size or the speed alone
Fig. 1 The regimes it separates. What matters for this essay is the horizontal axis itself — the number, and the fact that it has a length in it that somebody chose.

Where the group comes from

Take the momentum equation for an incompressible fluid. It has an inertia term, a pressure term and a viscous term.

Now measure everything in units natural to the problem: lengths in terms of some length LL of the body, velocities in terms of the free-stream speed UU, and time in terms of L/UL/U. Substitute those in and every term acquires a factor.

What emerges is that the equation can be written with one dimensionless coefficient in front of the viscous term, and that coefficient is

μρUL=1Re\frac{\mu}{\rho U L} = \frac{1}{Re}

Everything else cancels. So a flow’s behaviour cannot depend separately on the density, the viscosity, the speed and the size — it can depend only on that combination, because that is the only place they survive in the equation.

That is a much stronger statement than “the Reynolds number is a useful ratio”. It says that two flows with the same Reynolds number and the same geometry are governed by identical equations, and therefore have identical solutions.

The length nobody agrees on

Here is the difficulty. The derivation above needed a length LL, and the equations do not say which.

Any length characteristic of the problem will do, and different conventions have grown up in different corners of the subject:

A cylinder or sphere — the diameter.

A wing — the chord, measured along the flow direction.

A pipe — the internal diameter, or for non-circular ducts the hydraulic diameter, four times the area divided by the perimeter.

A flat plate — the distance from the leading edge, which makes the Reynolds number a function of position rather than a single number for the flow.

A boundary layer — sometimes the layer’s own thickness, or its momentum thickness, which gives numbers thousands of times smaller than the chord-based value for the same flow.

None is more correct than another. But a Reynolds number quoted without its length is a number without a meaning, and comparing two that used different conventions is the commonest way to misuse the whole idea.

What a threshold actually means

The consequence for the thresholds on the axis is worth spelling out.

“Vortex shedding begins at Reynolds 47” is shorthand for: for a circular cylinder in a uniform stream, with the Reynolds number based on the diameter, the steady wake becomes unstable near 47. Change the body and the number changes. Change the length convention and it changes. Change the free stream’s turbulence level and it moves a little.

“Pipe flow becomes turbulent at Reynolds 2300” is shorthand for the diameter-based value in a smooth circular pipe with a well-behaved entrance — and careful experiments have kept pipe flow laminar past 100,000 by eliminating disturbances.

So a threshold is a statement about a specific configuration, and quoting one to three significant figures without its conditions is precision that is not there.

Flow past a cylinder at Re 40A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.separatedrecirculation 0.56 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 40
Fig. 2 A flow at a stated Reynolds number, which on this site always means the cylinder’s diameter. The convention is on every figure for exactly the reason above.

Worked numbers

Some values, because the abstraction becomes concrete surprisingly fast.

A bacterium, 2 micrometres long at 30 micrometres a second in water: Re≈6×10−5Re \approx 6 \times 10^{-5}. Deep in the creeping regime, where nothing coasts and everything is reversible.

A raindrop, 2 millimetres across at 6 metres a second in air: Re≈800Re \approx 800. Separated, with a wake, and the drop is deformed by the pressure distribution.

A cyclist, 0.5 metres across at 10 metres a second: Re≈3×105Re \approx 3 \times 10^5. Right at the drag crisis, which is why clothing texture and helmet shape make a measurable difference.

An airliner wing, 4 metre chord at 250 metres a second at altitude: Re≈2×107Re \approx 2 \times 10^7. Boundary layer thin, turbulent over most of the chord, and ideal flow plus a layer correction is an excellent description.

A whale, 30 metres at 10 metres a second: Re≈3×108Re \approx 3 \times 10^8.

The span from bacterium to whale is thirteen orders of magnitude, and it is not a span of sizes — it is a span of regimes. Nothing that is true of the whale’s flow is true of the bacterium’s.

A model that is not a small aircraft

The consequence for testing is worth making concrete, because it is the most common practical trap.

A one-tenth-scale model of an aircraft, tested at the same speed, has one-tenth the Reynolds number. That is not a small change: it can move the flow from turbulent-attached to laminar-separated, which changes the drag by a large factor and can change the stall behaviour completely.

So a model at the wrong Reynolds number does not give slightly wrong answers; it can give answers to a different question. Model aircraft, for the same reason, are not scaled aircraft — their boundary layers are proportionally much thicker, their sections behave differently, and aerofoils designed for full size often perform poorly at model scale.

The escapes are to raise the speed, raise the density by pressurising the tunnel, or lower the viscosity by chilling the gas. All three are used, and all three are expensive, which is a fair indication of how seriously the problem is taken.

What the solver computed

The Reynolds number on every viscous figure here is based on the cylinder’s diameter, and the site says so on the figure rather than assuming a reader shares the convention.

Inside lib/flow.js it enters exactly where the derivation says it should: the kinematic viscosity in the diffusion term is set to UD/ReU D / Re with DD the body diameter, and it appears nowhere else. That is the whole of its influence on the computation, which is a direct demonstration of the claim above — one number, one place in the equations.

The transition it can demonstrate is measured rather than assumed. At Reynolds 1 and 10 the computed field has no reverse flow behind the body; at 40 and 100 it does, and the recirculating region grows. Nothing tells the solver which regime to produce.

The threshold values printed on the regime axis are quoted from the literature and the site says so. They are experimental facts about configurations this solver has not reproduced, and presenting them as outputs would be dishonest.

Non-dimensionalising as a habit

The technique that produced the Reynolds number is worth having as a habit, because it works far beyond fluids.

Write the governing equation. Choose scales natural to the problem for each variable. Substitute, and collect the coefficients. Whatever dimensionless groups survive are the only things the answer can depend on.

Doing that here reduces four independent quantities — density, viscosity, speed, size — to one. That is not a small saving: a study that would have needed a four-dimensional sweep of parameters needs a one-dimensional one, and results measured on a model transfer directly to full size.

The same procedure applied to a compressible flow produces the Mach number as well; applied to a free surface it produces the Froude number; applied to a rotating system it produces the Rossby number. Each time, the group emerges rather than being invented.

Why the equation-based view matters

The derivation above is not just a tidier presentation. It answers a question that the ratio-of-forces story cannot.

The ratio story says the Reynolds number compares inertia with viscosity, which is true and slightly vague — inertia where, viscosity of what. The equation view says something exact: it is the coefficient the viscous term acquires when the equation is written in the problem’s own units, and it is the only parameter in the equation.

That is why a high Reynolds number does not mean viscosity is unimportant. It means the viscous term has a small coefficient — and a term with a small coefficient can still dominate where its derivatives are large, which is exactly what happens in the boundary layer.

A small parameter multiplying the highest derivative is the signature of a singular perturbation, and recognising that is what turned the paradox into a solved problem.

The one place it is not enough

A caution that follows directly from the derivation.

The derivation showed that for an incompressible flow, one group survives. The words “for an incompressible flow” were doing work: they removed the energy equation and fixed the density, and with them went every other group that could have appeared.

Put them back and more groups emerge. Allow density to vary and the Mach number appears — and above about a third of the speed of sound it starts to matter. Allow a free surface and gravity gives the Froude number. Allow surface tension and the Weber number.

Each additional group is another dimension of parameter space that a test has to match, and matching two at once in one facility is usually impossible.

Ship testing is the standard example, and its solution is instructive: the model is run at the correct Froude number so the wave pattern is right, the resulting Reynolds number is badly wrong, and the drag is split into a wave part scaled by Froude and a friction part estimated separately by a correlation. Two rules, one experiment, added back together.

Mach number: one number, four different flowsMach number is speed ÷ speed of sound. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.incompressiblecompressible, subsonictransonicsupersonica cyclistdensity starts to matteran airliner cruisingshock waves everywhereMach numberspeed ÷ speed of soundMachthe ratio decides the regime, not the size or the speed alone
Fig. 3 The second group, and the threshold where it starts to matter. Below about Mach 0.3 the Reynolds number is the whole story; above it, there are two numbers to match and one facility.
The same shape, two different worldsA sphere at Reynolds number a ten-thousandth and a sphere at a hundred thousand are not the same problem at different speeds. In one, motion stops the instant the forcing does; in the other, the object drags a wake behind it for many diameters.a bacteriumRe 10⁻⁴everything reverses if it stopsa thrown ballRe 10⁵a wake it drags behind itschematic — the contrast, not a solved fieldtwo regimes
Fig. 4 And what the first group alone already decides. These two are separated by Reynolds number and by nothing else, and they are not the same problem.

Roughness is a length too

A consequence of geometric similarity that catches people, and it is the reason model tests can mislead even when the Reynolds number is matched.

Similarity requires the two flows to be geometrically identical, and “geometrically” includes the surface. A one-tenth-scale model with the same surface finish as the full-size article is proportionally ten times rougher, and roughness is what decides whether a boundary layer stays laminar or trips.

So a model polished to the same standard as the aircraft is not a scaled model. To be similar its roughness must scale down as well, which for a small model means a finish finer than any practical manufacturing process.

In practice the effect is often exploited rather than fought: models are deliberately tripped with a band of roughness near the leading edge, forcing transition at a known station so that the layer over the rest of the model is turbulent as it would be at full scale. The model is then wrong in a controlled way instead of an uncontrolled one, which is the best available bargain.

Where the model stops

The length is a convention and must be stated.

One number is not always enough. Compressibility, gravity or surface tension each bring their own group, and matching only Reynolds is then insufficient.

Thresholds are configuration-specific and sensitive to disturbances.

Geometric similarity is assumed. Two flows with the same Reynolds number are identical only if the shapes are too — including surface roughness, which is a length and therefore scales.

Local Reynolds numbers

A refinement that resolves a confusion, and it is worth having because it is where the flat-plate case becomes intelligible.

On a flat plate there is no single length. The natural choice is distance from the leading edge, which makes the Reynolds number grow along the surface: small near the front, large further back.

That is not an awkwardness; it is the physics. The boundary layer at the leading edge really is in a different regime from the layer a metre downstream, and the transition from laminar to turbulent happens at a station where the local value passes a threshold — around 5×1055 \times 10^5 for a smooth plate in a quiet stream.

So a single flow can span regimes along its own length, and a body with a leading edge always does. Near enough to any leading edge the local Reynolds number is small, viscosity dominates, and the thin-layer approximation does not yet apply.

That is why boundary-layer solutions are singular at the leading edge, and why numerical work there needs care.

What it does not measure

Three things it is regularly asked to indicate and does not.

It is not a measure of turbulence. A high Reynolds number makes turbulence possible, not certain. A carefully disturbed pipe goes turbulent at 2,000; a carefully undisturbed one has been kept laminar past 100,000. The number sets the stage and the disturbances decide.

It is not a measure of speed. Two flows at the same Reynolds number can differ in speed by orders of magnitude, which is the entire basis of model testing.

It is not a measure of difficulty. Creeping flow at Reynolds 10−410^{-4} is analytically tractable and often has closed-form solutions; flow at Reynolds 100 is neither. Difficulty peaks in the middle, where neither limit is available.

That last one is worth remembering when a large Reynolds number is treated as intimidating. The extremes are the easy cases; it is the intermediate range that requires computation.

Who measured it, and when

The group appears in Stokes’s work in 1851. Osborne Reynolds gave it experimental meaning in 1883 with an apparatus of memorable simplicity: a glass tube fed from a tank, and a filament of dye introduced on the axis.

At low flow the filament drew a straight line the whole length of the tube. Turned up, at a definite point it broke into eddies and mixed. Reynolds varied the tube diameter and the fluid’s viscosity, and found the transition happened at the same value of his group each time — which is what established that the group, and not any of its ingredients, was what mattered.

He also noticed that the transition value depended on how carefully the inlet was arranged, which is the observation later work on disturbance sensitivity is built on. It is in the original paper.

The number on every figure here

A note on this site’s own practice, since the essay has argued that a Reynolds number without its convention is meaningless.

Every viscous figure carries its value in the corner, and the convention is the cylinder’s diameter throughout. That is stated in the model note rather than left implicit, and it is why those notes exist.

The ideal-flow figures carry no Reynolds number at all — and that absence is informative rather than an omission. Ideal flow has no Reynolds number in its equations, which is precisely the statement that viscosity has been removed. A figure that reported one would be claiming a dependence the model does not have.

So the presence or absence of the number on a figure here says which theory produced it, before reading the caption.

Flow past a cylinder at Re 100A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.separatedrecirculation 1.16 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 100
Fig. 5 A viscous figure, and the number is part of the claim: this is what happens at Reynolds 100 and not in general.
Ideal flow past a cylinderA uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.ideal flow — inviscid, irrotational, steadyno circulation
Fig. 6 An ideal one, with no Reynolds number because there is none to state. The same picture describes every speed and every fluid, which is both its strength and exactly what is wrong with it.

The ladder from here

Nearby: non-dimensionalisation as a general technique; the hydraulic diameter and duct flows; transition in pipes and its sensitivity to disturbance; and the boundary-layer Reynolds numbers based on thickness.

Then across to the regimes the number separates and to the layer whose thinness it governs.

Where this goes next

creepingattachedseparated, sheddingturbulentbacterium swimmingshedding begins, Re ≈ 47a thrown ballan airliner winga whaleReynolds numberinertia ÷ viscositylog₁₀ Rethe ratio decides the regime, not the size or the speed alone One number decides which physics applies A bacterium and a whale both swim, and they are not doing the same thing at different sizes. The ratio of inertia to viscosity separates them, and crossing it changes the rules rather than the magnitudes. separatedrecirculation 0.56 Dviscous flow, solved on a coarse grid — the bubble is under-resolvedRe = 40 Everything happens in a layer you cannot see Air has so little viscosity that ignoring it works almost everywhere. Almost everywhere leaves out a film next to the surface, perhaps a millimetre thick, and that film decides drag, stall and whether an aircraft flies at all. 0.871.330.85ideal flow — incompressible, so the tube's area sets the speed Mass has nowhere to go Squeeze a stream of fluid and it speeds up, not because anything pushes it but because the same amount has to get through a smaller gap every second. Almost every result in the subject is that observation with more machinery attached.