Reynolds Number: Laminar vs Turbulent Flow

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The concept
Why it matters
Friction factor
Assumptions
Typical Reynolds numbers

In 1883 Osborne Reynolds injected a thin stream of dye into water flowing through a glass tube. At low speeds the dye stayed a neat line; above a certain speed it broke up and filled the tube. He found that the change didn’t depend on speed alone, but on one dimensionless number that now carries his name. The Reynolds number is the first thing to calculate in almost any flow problem, because it decides which formulas apply.

Reynolds' experiment. At low Reynolds number the dye injected into the flow stays a thin line, and the speed varies smoothly from zero at the wall to a maximum at the centre (a parabola). At high Reynolds number eddies scatter the dye across the bore, and the velocity profile is much flatter.

The concept

The Reynolds number compares two effects in a moving fluid: inertia, which keeps a disturbance going, and viscosity, which damps it out.

Re = \frac{\rho\,V\,D}{\mu} = \frac{V\,D}{\nu}

Symbol	Meaning	Unit
$Re$	Reynolds number	none
$\rho$	Fluid density	kg/m³
$V$	Average flow velocity	m/s
$D$	Characteristic length: the inside diameter for a round bore	m
$\mu$	Dynamic viscosity	Pa·s
$\nu$	Kinematic viscosity, $\mu / \rho$	m²/s

When viscosity wins (low Re), the fluid moves in smooth, parallel layers: laminar flow. When inertia wins (high Re), small disturbances grow into swirling eddies: turbulent flow.

Re (flow in a round bore)	Flow regime
Below about 2300	Laminar
About 2300 to 4000	Transitional: can switch between the two
Above about 4000	Turbulent

Other shapes have their own critical values. For flow along a flat plate, transition happens at a Reynolds number (based on distance from the leading edge) of about 500 000. For non-circular ducts, use the hydraulic diameter, $D_h = 4A/P$ (four times the area over the wetted perimeter).

Why it matters

Friction and pressure loss. Laminar losses grow in proportion to velocity; turbulent losses grow roughly with its square, and they depend on wall roughness.
Heat transfer. Turbulence mixes the fluid, so heat moves several times faster than in laminar flow. Heat exchangers are designed to run turbulent.
Mixing and noise. Turbulent flow mixes well but is noisier and causes vibration.
Scale models. Two flows with the same Reynolds number (and the same shape) behave alike, which is how wind tunnel tests on scale models predict full-size results.

Friction factor

The pressure lost to friction in a straight bore is given by the Darcy–Weisbach equation, $\Delta p = f\,(L/D)\,(\rho V^2/2)$ . The friction factor $f$ depends on the regime:

\text{Laminar: } f = \frac{64}{Re}, \qquad \text{Turbulent (Haaland): } \frac{1}{\sqrt f} = -1.8\,\log_{10}\left[\left(\frac{\varepsilon/D}{3.7}\right)^{1.11} + \frac{6.9}{Re}\right]

where $\varepsilon$ is the wall roughness (about 0.0015 mm for drawn tubing and plastic, 0.045 mm for commercial steel, 0.26 mm for cast iron). The Haaland equation is an explicit form of the Colebrook equation, accurate to about 2 %, and it gives the same values you would read from a Moody chart.

Assumptions

Steady, fully developed flow (away from the entrance, bends and fittings).
A round, full bore with an incompressible Newtonian fluid.
Constant viscosity, so the fluid temperature doesn’t change much along the length.

Solved problem 1: oil in a hydraulic hose

ISO VG 46 hydraulic oil (ν = 46 × 10⁻⁶ m²/s, ρ = 870 kg/m³ at 40 °C) flows at 20 L/min through a hose with a 12 mm bore, 5 m long. Is the flow laminar or turbulent, and what is the pressure drop?

Step 1: velocity.

A = \frac{\pi \times 0.012^2}{4} = 1.131 \times 10^{-4}\ \text{m}^2, \qquad Q = \frac{20}{60\,000} = 3.33 \times 10^{-4}\ \text{m}^3/\text{s}, \qquad V = \frac{Q}{A} = 2.95\ \text{m/s}

Step 2: Reynolds number.

Re = \frac{V D}{\nu} = \frac{2.95 \times 0.012}{46 \times 10^{-6}} = 769

Well below 2300, so the flow is laminar.

Step 3: friction factor and pressure drop.

f = \frac{64}{769} = 0.0832, \qquad \Delta p = 0.0832 \times \frac{5}{0.012} \times \frac{870 \times 2.95^2}{2} = 131\,000\ \text{Pa}

AnswerLaminar (Re ≈ 770); pressure drop about 131 kPa (1.3 bar) over 5 m

Because the flow is laminar, the pressure drop is proportional to the viscosity. On a cold morning the oil can be ten times thicker, and the loss rises with it, which is why hydraulic systems are sized for the coldest oil they will see.

Solved problem 2: water in a steel line

Water at 20 °C (ν = 1.0 × 10⁻⁶ m²/s) flows at 1.5 m/s through a commercial steel bore of 50 mm (ε = 0.045 mm). Find the friction factor and the pressure drop over 100 m.

Step 1: Reynolds number.

Re = \frac{1.5 \times 0.050}{1.0 \times 10^{-6}} = 75\,000 \quad \text{(turbulent)}

Step 2: relative roughness.

\frac{\varepsilon}{D} = \frac{0.045}{50} = 0.0009

Step 3: friction factor from the Haaland equation.

\frac{1}{\sqrt f} = -1.8\,\log_{10}\left[\left(\frac{0.0009}{3.7}\right)^{1.11} + \frac{6.9}{75\,000}\right] = -1.8\,\log_{10}(1.89 \times 10^{-4}) = 6.70

f = \frac{1}{6.70^2} = 0.0223

Step 4: head loss and pressure drop.

h_f = 0.0223 \times \frac{100}{0.050} \times \frac{1.5^2}{2 \times 9.81} = 5.11\ \text{m}, \qquad \Delta p = 1000 \times 9.81 \times 5.11 = 50.1\ \text{kPa}

Answerf ≈ 0.022; pressure drop about 50 kPa (0.5 bar) per 100 m

Typical Reynolds numbers

Flow	Approximate Re
Oil in a bearing film	Below 1000 (laminar)
Hydraulic oil in a hose	Hundreds to a few thousand
Blood in a large artery	About 1000–4000
Water from a household tap	About 10 000–50 000
Air over a car at motorway speed (based on its length)	About 10 million
Air over an airliner wing	Tens of millions

Key points

Re = ρVD/μ compares inertia with viscosity.
In a round bore: laminar below about 2300, turbulent above about 4000.
Laminar friction factor is 64/Re; turbulent friction also depends on roughness (Moody chart, Colebrook or Haaland).
Turbulence raises pressure loss but also mixing and heat transfer.

Reynolds number and fluid flow: laminar vs turbulent

The concept

Why it matters

Friction factor

Assumptions

Typical Reynolds numbers

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