# Reynolds Number

## Introduction and definition of the dimensionless Reynolds Number - online calculators

The Reynolds Number, the non-dimensional velocity, can be defined as the ratio of

• the inertia force (ρ u L), and
• the viscous or friction force (μ)

and interpreted as the ratio of

• the dynamic pressure (ρ u2), and
• the shearing stress (μ u / L)

and can be expressed as

Re = (ρ u2) / (μ u / L)

= ρ u L / μ

= u L / ν                                                      (1)

where

Re = Reynolds Number (non-dimensional)

ρ = density (kg/m3, lbm/ft3  )

u = velocity based on the actual cross section area of the duct or pipe (m/s, ft/s)

μ = dynamic viscosity (Ns/m2, lbm/s ft)

L = characteristic length (m, ft)

ν = μ / ρ = kinematic viscosity (m2/s, ft2/s)

### Reynolds Number for a Pipe or Duct

For a pipe or duct the characteristic length is the hydraulic diameter.

L = dh

where

dh = hydraulic diameter (m, ft)

The Reynolds Number for a duct or pipe can be expressed as

Re = ρ u dh / μ

= u dh / ν                                              (2)

where

dh = hydraulic diameter (m, ft)

#### Reynolds Number for a Pipe or Duct in Imperial Units

The Reynolds number for a pipe or duct can also be expressed in Imperial units

Re = 7745.8 u dh / ν                                   (2a)

where

Re = Reynolds Number (non dimensional)

u = velocity (ft/s)

dh = hydraulic diameter (in)

ν = kinematic viscosity (cSt) (1 cSt = 10-6 m2/s )

The Reynolds Number can be used to determine if flow is laminar, transient or turbulent. The flow is

• laminar - when Re < 2300
• transient - when 2300 < Re < 4000
• turbulent - when Re > 4000

In practice laminar flow is only actual for viscous fluids - like crude oil, fuel oil and oils.

### Example - Calculating Reynolds Number

A Newtonian fluid with a dynamic or absolute viscosity of 0.38 Ns/m2 and a specific gravity of 0.91 flows through a 25 mm diameter pipe with a velocity of 2.6 m/s.

The density can be calculated using specific gravity like

ρ = 0.91 (1000 kg/m3)

= 910 kg/m3

The Reynolds Number can then be calculated using equation (1) like

Re = (910 kg/m3) (2.6 m/s) (25 mm) (10-3 m/mm) / (0.38 Ns/m2)

= 156 ((kg m / s2)/N)

= 156 ~ Laminar flow

1 (N) = 1 (kg m / s2)

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### Online Reynolds Calculator

#### Density and the absolute (dynamic) viscosity is known

The calculator below can be used if the density and the absolute (dynamic) viscosity of a fluid is known. The calculator is valid for incompressible flow - flow with fluids or gases without compression - as typical for an air flow in a HVAC systems or similar.

Density - ρ - (kg/m3, lbm/ft3)

Velocity - u - (m/s, ft/s)

Characteristic length - L - (hydraulic diameter - dh ) (m, ft)

Absolute (dynamic) viscosity - μ - (Ns/m2, lbm/s ft)

The default values are for air at 60 oF, 2 atm pressure and density 0.146 lbm/ft3, flowing 20 ft/s between two metal sheets with characteristic length 0.5 ft. Dynamic (absolute) viscosity is 1.22 10-5 lbm/s ft.

#### Kinematic viscosity is known

The calculator below can be used when the kinematic viscosity of a fluid is known.

Velocity - u - (m/s, ft/s)

Characteristic length - L - (hydraulic diameter - dh ) (m, ft)

Kinematic viscosity - ν - (m2/s, ft2/s)) (1 cSt = 10-6 m2/s)

The default values are for water at 20oC with kinematic viscosity 1.004 10-6 m2/s in a schedule 40 steel pipe. The characteristic length (hydraulic diameter) of the pipe is 0.102 m.

## Related Topics

• Fluid Mechanics - The study of fluids - liquids and gases. Involves velocity, pressure, density and temperature as functions of space and time
• Fluid Flow and Pressure Drop - Pipe lines - fluid flow and pressure loss - water, sewer, steel pipes, pvc pipes, copper tubes and more
• Water Systems - Hot and cold water service systems - design properties, capacities, sizing and more
• Piping Systems - Dimensions of pipes and tubes, materials and capacities, pressure drop calculations and charts, insulation and heat loss diagrams

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