What is Reynolds Number (Re)?

In fluid mechanics, the Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of forces for given flow conditions. Thus, it is used to identify different flow regimes, such as laminar or turbulent flow.

It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similar flow patterns, in perhaps different fluids with possibly different flowrates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar.

It is named after Osborne Reynolds (1842–1912), who proposed it in 1883.

In general, Reynolds Number is calculated as follows:

Re = {\rho v_s^2/L\over {\mu v_s/L^2}} = {\rho v_sL\over \mu} = {v_sL\over \nu} = {\hbox{Inertial forces}\over {\hbox{Viscous forces}}}


vs = mean fluid velocity
L = Length
μ = dynamic fluid viscosity
ν = kinematic fluid viscosity
ρ = fluid density

For a pipe, Reynolds Number can be calculated as follows:

Re \frac{VD_p}{\mu} = \frac{VD}{\nu}


V = average velocity (ft/sec)
D = inside pipe diameter (ft)
γ = kinematic viscosity (ft2/s)
υ = dynamic viscosity (lbf·sec/ft2)
ρ = mass density (slugs/ft3)

The difference between laminar and turbulent flow:

The transition between laminar and turbulent flow is often indicated by a critical Reynolds number which depends on the exact flow configuration and must be determine through experimentation. It is important for engineers to never design for this transition period, and therefore engineers will generally stay away from about the 2000 to 3000 range to ensure that the flow is either laminar or turbulent (for a circular pipe Retransition is around 2300).

Laminar Flow

Laminar flow exists when the fluid particles move along in smooth paths. The average velocity of such flow is relatively low and doesn't often occur in engineering applications. Laminar flow occurs when Re<2000 and in such cases, the friction factor is only a function of Re. The energy head lost is a result of the fluid viscosity and friction loss varies with the velocity.

Transitional Flow

Transitional flow occurs in a critical zone where the average velocity changes the flow form laminar to turbulent. Transitional flow occurs when 2000<Re<4000.

Turbulent Flow

Turbulent flow exists when the fluid particles move in very irregular paths. The energy head lost is a result of the turbulence and friction loss varies with square of the velocity. Turbulent flow occurs when 4000 < Re.

How to determine Reynolds number:

One way to determine Reynolds number is to use Moody's Diagram. Pressure drops seen for fully-developed flow of fluids through pipes can be predicted using the Moody diagram which plots the friction factor(f) against Reynolds number (Re) and relative roughness (ε / D). The diagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases.

Figure 1: Moody Diagram
Figure 1: Moody Diagram (Click to Enlarge)



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