Reynolds Number: Laminar and Turbulent Flow Explained

Smooth Flow and Chaotic Flow

Open a tap slightly and the water falls as a smooth, glassy column. Open it fully and the stream becomes rough and churning. Smoke from a candle rises in a smooth thread and then suddenly breaks into swirls. These are the two kinds of fluid flow: laminar and turbulent. Which one occurs is predicted by a single number, the Reynolds number. This tutorial explains laminar and turbulent flow, how to calculate the Reynolds number and interpret it, Poiseuille's law for flow through pipes, and applications from water supply and blood flow to aircraft and cars. It builds on the tutorials on viscosity and Bernoulli's principle.

Laminar Flow

In laminar (streamline) flow, the fluid moves in smooth, parallel layers that slide past one another without mixing. Every particle passing a given point follows the same path, so the flow can be drawn with steady streamlines that never cross.

In a pipe, laminar flow has a characteristic velocity profile. The fluid touching the wall is at rest (the "no-slip" condition), and the speed increases smoothly towards the centre, where it is greatest. The profile is a parabola, and the maximum speed at the centre is twice the average speed.

Laminar flow happens when the fluid is slow, the pipe is narrow, or the fluid is very viscous, as with honey poured from a spoon or oil in a thin bearing.

Turbulent Flow

In turbulent flow, the motion is irregular and chaotic. Eddies and swirls form, break up and re-form, and the fluid mixes rapidly. The velocity at any point fluctuates wildly in size and direction, although the average flow is still along the pipe.

Turbulent flow has important consequences:

much more energy is lost to friction, so pumps must work harder;
mixing and heat transfer are much more effective, which is useful in heat exchangers and for mixing fuel and air in engines;
it is noisy: a narrowed artery can make turbulent blood flow that a doctor hears through a stethoscope.

The Reynolds Number

Whether a flow is laminar or turbulent depends on a competition between the fluid's inertia (its tendency to keep moving and form eddies) and its viscosity (internal friction that damps eddies out). The ratio of these is the Reynolds number:

Re = ρvd / η

ρ = density of the fluid (kg/m³)
v = mean flow speed (m/s)
d = a characteristic length, the diameter for a pipe (m)
η = viscosity of the fluid (Pa s)

The Reynolds number has no units: the units of ρvd/η cancel completely. It is a pure number, which is what makes it so useful: two flows with the same Re behave in the same way, whatever their size.

For flow in a circular pipe, typical guidelines are:

Re below about 2,000: laminar flow.
Re between about 2,000 and 4,000: transitional, with flow switching between the two.
Re above about 4,000: turbulent flow.

The exact boundaries depend on how smooth the pipe is and how disturbed the flow is at the inlet.

Calculating the Reynolds Number: Worked Examples

Example 1. Water (ρ = 1000 kg/m³, η = 1.0 × 10⁻³ Pa s) flows at 0.5 m/s through a pipe 2.0 cm in diameter. Is the flow laminar or turbulent?

Re = 1000 × 0.5 × 0.020 / 1.0 × 10⁻³ = 10,000. This is well above 4,000, so the flow is turbulent. Most water flowing in household pipes is turbulent.

Example 2. What is the greatest mean speed at which water can flow through the same pipe and remain laminar?

Set Re = 2000: v = Re × η/(ρd) = 2000 × 1.0 × 10⁻³ / (1000 × 0.020) = 0.10 m/s. Water must flow very slowly for laminar flow in an ordinary pipe.

Example 3. Blood (ρ = 1060 kg/m³, η = 3.5 × 10⁻³ Pa s) flows through the aorta (diameter 2.5 cm) at an average speed of 0.30 m/s.

Re = 1060 × 0.30 × 0.025 / 3.5 × 10⁻³ ≈ 2,270. This is in the transitional range. During the peak of each heartbeat, when the blood moves faster, the flow in the aorta can briefly become turbulent. In the small vessels, with tiny diameters and slow speeds, Re is far below 1 and the flow is strongly laminar.

Example 4. Honey (ρ = 1400 kg/m³, η = 10 Pa s) flows at 0.1 m/s through a 1 cm tube: Re = 1400 × 0.1 × 0.01 / 10 = 0.14, extremely laminar. Its huge viscosity crushes any eddies.

Poiseuille's Law for Laminar Flow

For laminar flow through a pipe of radius r and length L, driven by a pressure difference ΔP, the volume flow rate Q (m³/s) is given by Poiseuille's law:

Q = πr⁴ΔP / (8ηL).

The striking feature is the fourth power of the radius. Halving the radius divides the flow by 2⁴ = 16, for the same pressure difference.

Example 5. Water flows through a tube of radius 1.0 mm and length 0.50 m under a pressure difference of 2,000 Pa. Find the flow rate.

Q = π × (1.0 × 10⁻³)⁴ × 2000 / (8 × 1.0 × 10⁻³ × 0.50) = 1.57 × 10⁻⁶ m³/s, about 94 mL per minute.

The mean speed is Q/(πr²) = 0.50 m/s. Check that the law applies: Re = 1000 × 0.50 × 0.002 / 1.0 × 10⁻³ = 1,000, below 2,000, so the flow is laminar and Poiseuille's law is valid.

Why it matters in medicine. If fatty deposits narrow an artery to half its radius, the heart would need 16 times the pressure difference to maintain the same blood flow. In reality, flow falls and the tissue beyond receives too little blood. Small changes in blood vessel radius are also how the body controls blood flow to different organs.

Applications of the Reynolds Number

Model testing: engineers test small models of ships, aircraft and buildings in tanks and wind tunnels. For the model to behave like the real thing, the Reynolds numbers must match, which often means using faster flow or a different fluid for the model.
Golf balls: the dimples deliberately trigger turbulence in the thin layer of air next to the ball. A turbulent boundary layer clings to the ball longer, leaving a smaller wake and less drag, so a dimpled ball flies about twice as far as a smooth one.
Water supply and pipelines: engineers use the Reynolds number to choose pipe sizes and pump powers, since turbulent flow needs much more pumping energy.
Microorganisms: bacteria swim at Reynolds numbers far below 1. For them, water is like thick syrup: when they stop swimming they stop almost instantly, with no gliding. They need corkscrew-like flagella rather than flapping strokes.

Summary

Laminar flow is smooth and layered; turbulent flow is chaotic with eddies and rapid mixing.
Reynolds number Re = ρvd/η is dimensionless and compares inertia with viscosity.
In pipes, Re below about 2,000 gives laminar flow and above about 4,000 gives turbulent flow.
Poiseuille's law, Q = πr⁴ΔP/(8ηL), applies to laminar flow; flow depends on the fourth power of the radius.
Flows with the same Reynolds number behave alike, which makes scale-model testing possible.