Reynolds Number — Physica

Equivalent forms

Re = \frac{v L}{\nu}

A single dimensionless number predicts the entire flow regime.

Symbol	Name	SI unit	Dimension	Range
$Re$	Reynolds Numberoutput Dimensionless flow regime indicator	dimensionless	1	0 – 1000000
$ρ$	Density Fluid density	kg/m³	M·L⁻³	1 – 13600
$v$	Velocity Characteristic flow velocity	m/s	L·T⁻¹	0 – 50
$L$	Characteristic Length Characteristic length scale (e.g., pipe diameter)	m	L	0.0001 – 10
$μ$	Dynamic Viscosity Fluid dynamic viscosity	Pa·s	M·L⁻¹·T⁻¹	0.00001 – 10

Unit systems

Symbol	SI	CGS	Imperial
$Re$	dimensionless	dimensionless	dimensionless
$ρ$	kg/m³	g/cm³	lb/ft³
$v$	m/s	cm/s	ft/s
$L$	m	cm	ft
$μ$	Pa·s	P	lb/(ft·s)

Where it holds

Newtonian fluids; transition Re depends strongly on geometry

(\approx 2300

for pipes,

\approx 5\times 10^{5} for

flat plates).

Dimensional analysis

$[\rho vL/\mu ] = (M\cdot L^{-3})(L\cdot T^{-1})(L) / (M\cdot L^{-1}\cdot T^{-1}) = (M\cdot L^{-1}\cdot T^{-1})/(M\cdot L^{-1}\cdot T^{-1}) = 1 \checkmark$ (dimensionless)

Discovery

Osborne Reynolds · 1883

Reynolds injected dye into pipe flows and observed the transition from laminar streaks to turbulent swirls, identifying the dimensionless number that governs the transition.

Try this

When does smooth flowing water become a chaotic turbulent mess?

Water (ρ=1000 kg/m³, μ=0.001 Pa·s) flows through a 5 cm pipe at 1 m/s. Find Re and determine if the flow is laminar or turbulent.

Research status: stable

Real-world applications

Pipe and duct design in plumbing and HVAC
Aircraft and ship hydrodynamics scaling (wind tunnel models)
Microfluidic device design (low-Re regime)
Blood flow analysis in arteries
Predicting drag coefficients on spheres and cylinders

Common misconceptions

$Re = 2300$ is NOT a universal turbulence threshold — it's specific to pipe flow
Re depends on the chosen characteristic length L — different choices give different numbers for the same flow
High Re does NOT mean 'fast' — it means 'inertially dominated', which can occur at any speed $if \mu is$ small enough

Experimental verification

Reynolds's original 1883 dye-injection experiment in glass pipes showed sharp laminar-to-turbulent transition near

Re \approx 2000

. Repeated countless times since with hot-wire anemometry, PIV, and DNS simulations.

Derivation

Re comes from non-dimensionalizing the Navier-Stokes equation.

The inertial term

\rho v\cdot \nabla v

scales

as \rho v^{2}/L

, and the viscous term

\mu \nabla ^{2}v

scales

as \mu v/L^{2}

.

Their ratio is

(\rho v^{2}/L)/(\mu v/L^{2}) = \rho vL/\mu \equiv Re

.

When Re ≫ 1, inertia dominates; when Re ≪ 1, viscosity dominates.

Limiting cases

Re < 2300 (pipe)⟶ Laminar flowSmooth, layered streamlines — viscous forces dominate.

2300 < Re < 4000⟶ TransitionalIntermittent turbulent bursts — unpredictable regime.

Re > 4000⟶ Turbulent flowChaotic eddies, enhanced mixing — inertial forces dominate.

Re \to 0

⟶ Stokes (creeping) flowViscosity completely dominates — used for microfluidics and bacteria swimming.

What if…

What if you doubled the pipe diameter at fixed velocity?

Re doubles. Larger pipes are more prone to turbulence at the same speed — which is why big rivers turbulate easily while capillaries stay laminar.

What if you replaced water with honey

(1000\times

more viscous)?

Re drops by $1000\times$ . Honey at 1 m/s in a 5 cm pipe has $Re \approx 50$ — well below the turbulence threshold, so it flows in smooth layers.

What if you scale a model airplane down

100\times

?

To preserve Re, you must increase $v\times L$ by 100. Wind-tunnel testing uses high-speed flow or pressurized gas to match the full-scale Re — otherwise the flow physics differs.

1

Water in a 5 cm pipe at 1 m/s

Given ·

ρ:: 1000
v:: 1
L:: 0.05
μ:: 0.001

Find · Re

Steps

$Re = \rho vL/\mu$
$Re = (1000)(1)(0.05) / 0.001$
$Re = 50 / 0.001 = 50000$ ≫ $4000 \to$ turbulent

Result ·

Re = (1000 \times 1 \times 0.05) / 0.001 = 50000

→ fully turbulent

2

Bacterium swimming in water

Given ·

ρ:: 1000
v:: 0.00003
L:: 0.000002
μ:: 0.001

Find · Re

Steps

$v \approx 30$ µm/s, $L \approx 2$ µm
$Re = (1000)(3\times 10^{-5})(2\times 10^{-6})/(0.001)$
$Re \approx 6\times 10^{-5}$ — viscosity dominates utterly

Result ·

Re = (1000)(3e-5)(2e-6) / 0.001 = 6e-5

→ deeply Stokes regime

Bernoulli's Equation→Continuity Equation→