Reynolds number

Reynolds number (Re) is a dimensionless quantity that helps predict fluid flow patterns in various situations. It measures the ratio of inertial to viscous forces.

Overview

At low Reynolds numbers, flow tends to be dominated by laminar (sheet-like) behavior. At high Reynolds numbers, turbulence dominates. The transition is driven by flow velocity, characteristic length, and fluid properties.

Reynolds numbers are useful for predicting transitions between laminar and turbulent flows in:

• Pipes and ducts
• Open channels
• Airflow over aircraft wings
• Blood vessels
• Industrial mixing

They help scale experiments from model to real-world applications.

Laminar flow (left) to turbulent flow (right) from a tap as Re increases.

The transition from laminar to turbulent flow visualized in open water.

Definition

The Reynolds number is defined as:

$Re={\frac {uL}{\nu }}={\frac {\rho uL}{\mu }}$

Where:

• $u$: Flow speed (m/s)
• $L$: Characteristic length (m)
• $\nu$: Kinematic viscosity (m²/s)
• $\rho$: Fluid density (kg/m³)
• $\mu$: Dynamic viscosity (Pa·s)

Images

The plume from a candle flame transitions from laminar to turbulent.

A vortex street around a cylinder. Forms between Re ~40 and 1000.

George Stokes introduced concepts leading to Reynolds number.

Osborne Reynolds popularized its use in 1883.

Applications

Re is used in fluid mechanics to characterize flow regimes:

• **Laminar flow**: Re < ~2300
• **Transitional flow**: ~2300 < Re < ~2900
• **Turbulent flow**: Re > ~2900

In a Pipe

$Re={\frac {uD_{H}}{\nu }}={\frac {\rho uD_{H}}{\mu }}$

Where:

• $D_{H}$: Hydraulic diameter
• $Q$: Volumetric flow rate
• $A$: Cross-sectional area
• $W$: Mass flow rate

Hydraulic Diameter

$D_{H}={\frac {4A}{P}}$

• For circular pipes: $D_{H}=D$
• For annular ducts: $D_{H}=D_{o}-D_{i}$

Derivation via Navier-Stokes

Start from: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{f}}

Non-dimensionalizing: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D\mathbf{v}}{Dt} = -\nabla p + \frac{1}{Re} \nabla^2 \mathbf{v} + \mathbf{f}}

Shows how viscosity vanishes as Re approaches infinity (inviscid behavior).

History

Reynolds's 1883 experiment with dyed water flow.

Diagram showing turbulent onset.

Osborne Reynolds used dyed water in a glass pipe to show laminar-turbulent transitions. At low velocities, the stream was smooth. At higher velocities, turbulence formed.

Flow Around Objects

Sphere

Creeping flow past a sphere.

Stokes' law applies at low Re: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_d = 6 \pi \mu r v}

Packed Beds

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Re = \frac{\rho v_s D}{\mu (1 - \varepsilon)}}

Where:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_s} : Superficial velocity
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} : Void fraction

Stirred Vessels

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Re = \frac{\rho N D^2}{\mu} = \frac{\rho V D}{\mu}} Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is rotation speed.

Flow Similarity

Flow patterns with similar Re match in model and full scale.

For dynamic similarity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Re}_m = \mathrm{Re}, \quad \mathrm{Eu}_m = \mathrm{Eu}}

Pipe Friction

Moody diagram: friction factor vs. Re and pipe roughness.

Used to predict pressure drops:

• Laminar: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = \frac{64}{Re}}
• Turbulent: Varies with Re and roughness

Critical Values

• Onset of turbulence in pipe: Re ~ 2300
• Golf ball dimples reduce drag by manipulating boundary layer (higher Re)

Real-World Values

• Amoeba: ~1×10⁻⁶
• Bacteria: ~1×10⁻⁴
• Human swimming: ~10⁶
• Whale: ~10⁸
• Large ship: ~10⁹
• Cyclone: ~10¹²

Related Concepts

• Prandtl number
• Péclet number
• Kinematic viscosity
• Laminar-turbulent transition

External Links

• Feynman on Reynolds Number
• Britannica Definition
• Reynolds Calculator