Dimensionless Numbers and Turbulence

Light a match, blow it out, and watch the smoke. For the first few centimetres it rises in a glassy, ruler-straight thread — utterly smooth. Then, quite suddenly and always at about the same height, that thread buckles, curls, and shatters into a churning, unpredictable mess. Nothing about the smoke changed; the air is the same, the heat is the same. So what tripped the switch? The answer is a single pure number — no units, no dimensions — that decides, for every flow in the universe, whether it will glide smoothly or dissolve into chaos. It is called the Reynolds number, and this page is about that one number.

The same number explains two things that sound unrelated. Why can an engineer trust a 1{:}50 scale model of an aeroplane, a few feet long, bolted in a wind tunnel, to predict how the real forty-metre jet will fly? And why does a swimming bacterium, a thousandth of a millimetre across, live in what feels to it like honey — a world where coasting is impossible and stopping is instantaneous? Both puzzles have the same one-word answer. Everything below is built from the ratio at the heart of the Navier–Stokes equations.

A tug-of-war: inertia versus viscosity

Every flowing fluid is caught in a contest between two tendencies. Inertia wants to keep the fluid barrelling along in whatever direction it is already going — it is the fluid's unwillingness to be turned or stopped. Viscosity is internal friction: neighbouring layers of fluid dragging on one another, smoothing out differences in velocity, dissipating motion into heat. Inertia makes flows wild and free-running; viscosity makes them sluggish, orderly, and obedient.

Osborne Reynolds' insight (1883) was that what matters is not the strength of either force alone but their ratio. Estimate each with the fluid's density \rho, a typical speed u, a characteristic length L (pipe diameter, wing chord, body size), and the dynamic viscosity \mu. The inertial force per unit volume scales like \rho u^2 / L, and the viscous force like \mu u / L^2. Take the quotient and the messy bits cancel:

\mathrm{Re} \;=\; \frac{\text{inertial force}}{\text{viscous force}} \;\sim\; \frac{\rho u^2/L}{\mu u/L^2} \;=\; \frac{\rho u L}{\mu} \;=\; \frac{u L}{\nu},

where \nu = \mu/\rho is the kinematic viscosity. The result is dimensionless — a bare number. Check: \rho is \text{kg/m}^3, u is \text{m/s}, L is \text{m}, and \mu is \text{kg/(m·s)}; multiply and divide and every unit cancels. A big \mathrm{Re} means inertia is winning (the flow runs free and, past a threshold, turbulent); a small \mathrm{Re} means viscosity is winning (the flow is gooey, orderly, and reversible).

Where the number comes from: non-dimensionalising Navier–Stokes

The tug-of-war picture is a scaling estimate, but the Reynolds number is not a hand-wave — it drops straight out of the governing equation when you strip the units off. Start from the incompressible Navier–Stokes momentum equation,

\rho\left(\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}\right) = -\nabla p + \mu\,\nabla^2 \mathbf{u}.

Now measure everything in units natural to the problem: lengths in units of L, speeds in units of U, time in units of L/U, and pressure in units of \rho U^2. Writing the starred quantities for the dimensionless versions (\mathbf{x} = L\mathbf{x}^\ast, \mathbf{u} = U\mathbf{u}^\ast, and so on) and substituting, the whole equation collapses to

\frac{\partial \mathbf{u}^\ast}{\partial t^\ast} + (\mathbf{u}^\ast\cdot\nabla^\ast)\mathbf{u}^\ast = -\nabla^\ast p^\ast + \frac{1}{\mathrm{Re}}\,\nabla^{\ast 2}\mathbf{u}^\ast.

Look at what happened: every reference to \rho, \mu, U and L has been swept into a single coefficient, 1/\mathrm{Re}, sitting in front of the viscous term. That is the punchline of the whole subject: once you fix the shape of the boundaries, the entire behaviour of an incompressible flow is governed by one number. Two flows with the same geometry and the same \mathrm{Re} obey the identical dimensionless equation, so their solutions are the same picture.

Dynamic similarity: why the scale model works

That last observation is worth a fortune to engineers. If the dimensionless equation depends only on \mathrm{Re}, then a small model and a full-size machine will have geometrically identical flow patterns — same streamlines, same separation points, the same dimensionless drag and lift — provided they are the same shape and their Reynolds numbers match. This is the principle of dynamic similarity, and it is why wind-tunnel testing is trustworthy rather than a guess.

To match \mathrm{Re} on a smaller model you must compensate for the shrunk length by pushing something else up. Set the two Reynolds numbers equal:

\frac{\rho_1 u_1 L_1}{\mu_1} = \frac{\rho_2 u_2 L_2}{\mu_2}.

In the same fluid (\rho and \mu equal), this forces u_1 L_1 = u_2 L_2: halve the size and you must double the speed. That is exactly why wind tunnels run so fast, or use pressurised (denser) air, or even cold nitrogen — anything to hit the full-scale \mathrm{Re} on a model that fits in the room.

Definition. \mathrm{Re} = \dfrac{\rho u L}{\mu} = \dfrac{u L}{\nu}, a dimensionless number built from density \rho, speed u, length scale L, and viscosity \mu (with \nu = \mu/\rho).
What it measures. The ratio of inertial forces to viscous forces. Large \mathrm{Re} = inertia dominates; small \mathrm{Re} = viscosity dominates.
The one governing parameter. Non-dimensionalising Navier–Stokes leaves 1/\mathrm{Re} as the sole coefficient, so (for fixed geometry) it alone sets the flow.
Dynamic similarity. Same shape + same \mathrm{Re} = same dimensionless flow. This is what makes scale-model testing valid.
Transition. Below a critical value the flow is laminar; above it, turbulent. For flow in a circular pipe the transition sits around \mathrm{Re} \approx 2300.

The laminar → turbulent transition

Turn one knob — the speed, say, and hence \mathrm{Re} — and a flow marches through distinct regimes. The classic laboratory is flow past a circular cylinder (think of wind on a chimney, or Reynolds' own dye-in-a-pipe experiment). Reveal the figure below to watch the same cylinder at three increasing Reynolds numbers.

At low \mathrm{Re} the flow is laminar: smooth, layered, steady, and — crucially — predictable. Viscosity irons out any little disturbance before it can grow. As \mathrm{Re} climbs, viscosity can no longer damp the wobbles fast enough; the wake starts to oscillate and shed vortices. Push \mathrm{Re} higher still and the flow tips over into full turbulence. In a straight circular pipe this transition happens near a critical value:

\mathrm{Re}_{\text{crit}} = \frac{\rho\, u\, D}{\mu} \approx 2300 \quad(\text{pipe flow, diameter } D).

Below \sim 2300 a pipe flow will settle back to laminar even if you jostle it; above it, disturbances snowball. (The number is not razor-sharp — with an exquisitely smooth, quiet inlet, laminar flow has been coaxed to \mathrm{Re} of many tens of thousands — but 2300 is the practical threshold for ordinary plumbing.)

What turbulence actually is

"Turbulent" is not just a fancy word for "fast" or "messy". Turbulence is a specific, rich state of motion with a few defining features:

Chaotic and unpredictable. The flow is deterministic (Navier–Stokes still holds exactly) yet so sensitive to initial conditions that its detailed future is effectively unforecastable — the same reason weather is hard.
Multi-scale. Turbulence contains eddies of a huge range of sizes at once, swirls within swirls within swirls, from the size of the whole flow down to millimetres.
The energy cascade. Energy is fed in at the large scales (the big eddies), then handed down to ever-smaller eddies without much loss, until at the very smallest scales viscosity finally bites and turns the motion into heat. Lewis Fry Richardson's rhyme catches it: "Big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls, and so on to viscosity."
Efficient mixing and extra drag. All that churning stirs the fluid far faster than molecular diffusion could — great for mixing fuel and air, costly in the extra resistance it adds.

Turbulence is often called the last great unsolved problem of classical physics: we have the exact equation, and still cannot predict a turbulent flow in full from first principles. It all traces back to that one nonlinear convective term, (\mathbf{u}\cdot\nabla)\mathbf{u}, unleashed once \mathrm{Re} lets inertia off the leash.

Worked examples

Example 1 — water in a household pipe. Water (\rho = 1000\ \text{kg/m}^3, \mu = 1.0\times10^{-3}\ \text{Pa·s}) flows at u = 1\ \text{m/s} through a pipe of diameter L = D = 0.02\ \text{m}. Then

\mathrm{Re} = \frac{\rho u L}{\mu} = \frac{(1000)(1)(0.02)}{1.0\times10^{-3}} = 20{,}000.

Far above 2300 — so the tap water is firmly turbulent, which is why it hisses.

Example 2 — a bacterium swimming in honey. A bacterium (L \approx 1\ \mu\text{m} = 10^{-6}\ \text{m}) swims through water at u \approx 30\ \mu\text{m/s}, with \nu = 10^{-6}\ \text{m}^2/\text{s}. Then

\mathrm{Re} = \frac{uL}{\nu} = \frac{(3\times10^{-5})(10^{-6})}{10^{-6}} = 3\times10^{-5}.

A minuscule \mathrm{Re}: viscosity utterly dominates inertia. For the bacterium, water feels as thick as honey does to us — coasting is impossible (it halts within an atom's width of stopping its flagellum), and it must "corkscrew" rather than paddle to get anywhere. Life at low \mathrm{Re} is a genuinely different world.

Example 3 — a wind-tunnel model. A full-size wing section has chord L_1 = 2\ \text{m} and flies at u_1 = 60\ \text{m/s}. A 1{:}8 model (L_2 = 0.25\ \text{m}) is tested in the same air. To match \mathrm{Re} the model must run at

u_2 = u_1\,\frac{L_1}{L_2} = 60\times\frac{2}{0.25} = 480\ \text{m/s}.

That is above the speed of sound — a warning that pure geometric shrinking can push you into a different regime (here, compressibility), which is why real facilities cheat with dense or cold gas to hit \mathrm{Re} at a sane speed.

The other numbers, in brief

The Reynolds number is the headline act, but it has a supporting cast of dimensionless numbers, each the ratio of two effects, each ruling its own kind of flow. You meet them by the same recipe — non-dimensionalise, read off the coefficient:

Mach number \mathrm{Ma} = u/a (speed over the speed of sound a) — measures compressibility. Below \mathrm{Ma}\approx0.3 a gas behaves as effectively incompressible; near and above 1 you get shock waves.
Froude number \mathrm{Fr} = u/\sqrt{gL} (inertia over gravity) — governs free-surface flows: ship wakes, river rapids, the "hydraulic jump" in your kitchen sink.

But when the fluid is a liquid, or a slow gas, and you care whether it will be smooth or turbulent, \mathrm{Re} is the one that matters.

Exactly backwards — and it is the most common trip-up with this number. Viscosity \mu sits on the bottom of the fraction, so a large \mathrm{Re} means viscosity is comparatively weak — inertia is running the show. Thick, sticky fluids like honey, glycerine or cold tar flow at tiny Reynolds numbers, which is precisely why they ooze so smoothly and never go turbulent. Fast-moving air over a wing, by contrast, has a huge \mathrm{Re} even though air is barely viscous at all.

Two follow-on traps. First, a big \mathrm{Re} is not about size alone: it is the product \rho u L / \mu. A large slow ship and a small fast dart can share the same \mathrm{Re}. Second, "\mathrm{Re}" says nothing on its own until you name the length scale L you used — pipe diameter, wing chord, body length are all different L's, so always quote what L means.

Because resolving a turbulent flow means capturing every eddy, from the largest down to the tiny dissipative ones where viscosity acts — and the gap between those scales grows like \mathrm{Re}^{3/4} in each direction. A full three-dimensional simulation that resolves everything (a "direct numerical simulation") needs on the order of \mathrm{Re}^{9/4} grid points. For the flow over an airliner, \mathrm{Re} is around 10^{7}, which demands more grid points than any computer that will exist for decades. So engineers model the small scales statistically instead of resolving them — that is what all those turbulence models (k–\varepsilon, large-eddy simulation, and the rest) are for. The equation is known exactly; it is the arithmetic that defeats us.