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# Reynolds number

The Reynolds number is the most important dimensionless number in fluid dynamics providing a criterion for dynamic similarity. It is named after Osbourne Reynolds (1842-1912). Typically it is given as follows:

$Re = {\rho v_{s} L\over \eta}$

or

$Re = {v_{s} L\over \nu} \; .$

With:

• vs - mean fluid velocity,
• L - characteristic length (equal to diameter 2r if a cross-section is circular),
• η - (absolute) dynamic fluid viscosity,
• ν - kinematic fluid viscosity: ν = η / ρ,
• ρ - fluid density.

The Reynolds number is used for determing whether a flow is laminar or turbulent. Laminar flow within e.g. pipes will occur when the Reynolds number is below the critical Reynolds number of Recrit, pipe = 2300 (or practically Re > 3000) and turbulent flow when it is above 2300 where the Reynolds number is based on the pipe diameter and the mean velocity vs within the pipe. The value of 2300 has been determined experimentally and a certain range around this value is considered the transition region between laminar and turbulent flow. Please note, that the critical Reynolds number Recrit depends on the flow type and the definition of the Reynolds number.

### The similarity of flows

Two flows with equal Reynolds numbers and the same geometry are similar. For the flow in a model and the real flow holds in appropriate points:

$Re^{\star} = Re \; , \quad\quad {p^{\star}\over \rho^{\star} {v^{\star}}^{2}} = {p\over \rho v^{2}} \; .$

Quantities marked with * concern the flow around the model and the others the real flow. This is useful for experiments with reduced model in the water channel[?] or in the wind tunnel, where we get data for the real flows. Note that in compressible flow the Mach number must also be equal for the two flows to be similar.

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