Reynolds Number

An introduction and definition of the dimensionless Reynolds Number - with online calculators

The Reynolds Number is a non dimensional parameter defined by the ratio of

• dynamic pressure (ρ u²) and
• 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, lb_m/ft³  )

u = velocity (m/s, ft/s)

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

L = characteristic length (m, ft)

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

 Re = ρ u d_h / μ

    = u d_h / ν          (2)

where

d_h = hydraulic diameter (m, ft)

Reynolds Number for a Pipe or Duct in common Imperial Units

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

Re = 7745.8 u d_h / ν           (2a)

where

Re = Reynolds Number (non dimensional)

u = velocity (ft/s)

d_h = 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

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 the 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)

Online Reynolds Calculator

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 air flow in HVAC systems or similar.

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

Density - ρ - (kg/m³, lb_m/ft³)

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

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

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

The calculator below can be used when the kinematic viscosity of a fluid is known. The default values are for water at 60^oC with a kinematic viscosity of 1.13 10^-6 m²/s in a schedule 40 steel pipe. The characteristic length (hydraulic diameter) of the pipe is 0.102 m.

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

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

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

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