# Reynolds number

The Reynolds number is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. Where two similar objects in perhaps different fluids with possibly different flowrates have similar fluid flow around them, they are said to be dynamically similar.

It is named after Osborne Reynolds (1842-1912), who proposed it in 1883. Typically it is given as follows:

[itex] \mathit{Re} = {\rho v_{s} L\over \mu} [itex]

or

[itex] \mathit{Re} = {v_{s} L\over \nu} \; . [itex]

With:

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

The Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and is used for determining whether a flow will be laminar or turbulent. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow, on the other hand, occurs at high Reynolds numbers and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations.

The transition between laminar and turbulent flow is often indicated by a critical Reynolds number (Recrit), which depends on the exact flow configuration and must be determined experimentally. Within a certain range around this point there is a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and predictions of fluid behaviour can be difficult. For example, within circular pipes the critical Reynolds number is generally accepted to be 2300, where the Reynolds number is based on the pipe diameter and the mean velocity vs within the pipe, but engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 4000 to ensure that the flow is either laminar or turbulent.

## The similarity of flows

In order for two flows to be similar they must have the same geometry and equal Reynolds numbers. When comparing fluid behaviour at homologous points in a model and a full-scale flow, the following holds:

[itex] \mathit{Re}^{\star} = \mathit{Re} \; , \quad\quad {p^{\star}\over \rho^{\star} {v^{\star}}^{2}} = {p\over \rho v^{2}} \; , [itex]

where quantities marked with * concern the flow around the model and the others the real flow. This allows us to perform experiments with reduced models in water channels or wind tunnels, and correlate the data to the real flows. Note that true dynamic similarity may require matching other dimensionless numbers as well, such as the Mach number used in compressible flows, or the Froude number that governs free-surface flows. Some flows involve more dimensionless parameters than can be practically satisfied with the available apparatus and fluids (preferably air or water), so one is forced to decide which parameters are most important. This is why good experimental modelling requires a fair amount of experience and good judgement.

## Example on the importance of Reynolds number

If an aeroplane needs testing of its wing, one can make a scaled down small model of the wing and test the wing as table top model in the lab with the same Reynolds number the actual air plane is subjected to. The results of the lab model will be similar to that of the actual plane wing results. Thus we need not bring a plane into the lab to test it actually. This is the example of "dynamic" "similarity." This is what Reynolds number is all about.

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