Reynolds number

The Reynolds number $\mathrm{Re}$ is a dimensionless number that compares the relative strengths of inertia and viscosity in a flow. It puts the smooth motion of slowly stirred honey and the momentum-dominated motion of rapidly driven water on one scale.

A low $\mathrm{Re}$ means viscosity is relatively important; a high $\mathrm{Re}$ means inertia is relatively important. $\mathrm{Re}$ alone does not uniquely determine a flow state: geometry, inlet disturbances and surface roughness also matter.

Inertia is the tendency of fluid motion to persist. It becomes more prominent for faster flows, larger bodies and denser fluids.

Viscosity smooths differences in velocity between neighbouring fluid. It becomes relatively stronger for a fluid with large dynamic viscosity $\mu$ or for a flow through a very small gap. $\mathrm{Re}$ compares these effects on one scale.

Re ≪ 1

Viscosity dominates; the flow responds quickly to changes in forcing with little inertial overshoot.

Re ≫ 1

Inertia dominates the bulk flow, although viscosity remains essential near solid walls.

Using characteristic velocity $V$, length $L$, density $\rho$ and dynamic viscosity $\mu$ gives Equation (1) [1,3]. The equivalent form uses kinematic viscosity $\nu=\mu/\rho$.

The numerator $\rho VL$ represents the inertial scale and $\mu$ the viscous scale. Their units cancel, so $\mathrm{Re}$ is dimensionless.

The SI unit of dynamic viscosity $\mu$ is $\mathrm{Pa{\cdot}s}$ and that of kinematic viscosity $\nu$ is $\mathrm{m^2/s}$ [4]. A Reynolds number has no unit.

The numerical value of $\mathrm{Re}$ depends on the chosen characteristic scales. Always state the definition of $\mathrm{Re}$ together with the flow being studied.

For pipe flow, the standard choice is the cross-sectional mean velocity $V$ and internal diameter $D$, giving $\mathrm{Re}_D=\rho VD/\mu$. Other ducts use hydraulic diameter; external flows use a relevant body diameter, chord or other governing length.

Circular pipe

$V$: cross-sectional mean velocity; $L$ = $D$: internal diameter

Cylinder or sphere

$V$: free-stream velocity; $L$: diameter

Airfoil

$V$: free-stream velocity; $L$: chord

For flow in a long, smooth circular pipe, $\mathrm{Re}_D<2300$ is commonly treated as laminar, roughly 2300–4000 as transitional and $\mathrm{Re}_D>4000$ as turbulent [1].

These are not perfectly sharp physical boundaries. Inlet disturbances, pipe vibration, fittings and surface roughness can shift transition. Near the thresholds, engineering judgement should not rely on one number alone.

Water at 20 °C flows through a pipe of internal diameter $D=0.020\ \mathrm{m}$ at mean velocity $V=0.50\ \mathrm{m/s}$. Use density $\rho=998\ \mathrm{kg/m^3}$ and dynamic viscosity $\mu=1.002\times10^{-3}\ \mathrm{Pa{\cdot}s}$ to calculate $\mathrm{Re}_D$ and classify the flow.

Use internal diameter $D$ as the length and mean velocity $V$ as the velocity. Substitution in Equation (2) gives $\mathrm{Re}_D\approx9.96\times10^3$. Because it is well above 4000, the developed pipe flow is classified as turbulent.

Enter density $\rho$, cross-sectional mean velocity $V$, inside diameter $D$ and dynamic viscosity $\mu$ to calculate the pipe Reynolds number and a flow-regime guide. The initial values match the water example above.

The classification is a guide for circular-pipe flow. Use an appropriate characteristic length and transition criterion for other flows.

Matching $\mathrm{Re}$ between geometrically similar models and full-scale systems matches the relative strength of inertia and viscosity. It is one condition for dynamic similarity in wind-tunnel and towing-tank tests [3].

Other dimensionless groups may also matter: Froude number for free surfaces and Mach number for high-speed gas flow. When every group cannot be matched, prioritise the physics that controls the result of interest.

Before calculating, define fluid temperature, characteristic velocity and characteristic length in one line.

Using radius instead of diameter

Pipe Reynolds number conventionally uses internal diameter $D$.

Substituting flow rate for V

Volume flow rate $Q$ is not velocity; use mean velocity $V=Q/A$.

Mixing μ and ν

Use either $\mathrm{Re}=\rho VL/\mu$ or $\mathrm{Re}=VL/\nu$.

Treating high Re as zero viscosity

No-slip walls and boundary layers still depend fundamentally on viscosity.

Pipe friction, pressure loss, drag, heat and mass-transfer correlations, and model-test conditions commonly use $\mathrm{Re}$ as an input.

Next, compare laminar and turbulent flow to see how velocity profiles, mixing and pressure losses change.

Review density, dynamic viscosity and kinematic viscosity before calculating. In the Navier–Stokes equation, $\mathrm{Re}$ appears as the ratio of inertial to viscous terms.

Prerequisites

Density, pressure & viscosity; Navier–Stokes equation

Next

Laminar and turbulent flow; pipe velocity profiles; pipe friction loss