Reynolds number

The Reynolds number is the ratio of inertial forces (v_sρ) to viscous forces (μ/L) and is used for determining whether a flow will be laminar or turbulent. It is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. When 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 for flow through a pipe:

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

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

where:

v_s - 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.

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 (Re_crit), 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 v_s 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, have equal Reynolds numbers and Euler Numbers. When comparing fluid behaviour at homologous points in a model and a full-scale flow, the following holds:

\mathit{Re}^{\star} = \mathit{Re} \;

\mathit{Eu}^{\star} = \mathit{Eu} \;   \quad\quad     i.e.   \quad  {p^{\star}\over \rho^{\star} {v^{\star}}^{2}} = {p\over \rho v^{2}} \; ,

where quantities marked with * concern the flow around the model and the others the real flow. This allows engineers to perform experiments with reduced models in water channels or wind tunnels, and correlate the data to the real flows, saving on costs during experimentation and on lab time. 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 (for example air or water), so one is forced to decide which parameters are most important. For experimental flow modelling to be useful it requires a fair amount of experience and good judgement on the part of the engineer.

Reynolds number sets the smallest scales of turbulent motion

In a turbulent flow, there is a range of scales of the fluid motions, sometimes called eddies. A single packet of fluid moving with a bulk velocity is called an eddy. The size of the largest scales (eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke-stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number. As Reynolds number increases, smaller and smaller scales of the flow are visible. In the smoke-stack, the smoke may appear to have many very small bumps or eddies, in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in the flow. The higher the Reynolds number, the greater the range of scales.

What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important to the flow. With a low level of viscosity, the smallest scales of fluid motion are undamped -- there is not enough viscosity to dissipate their motions. In contrast, a low Reynolds number indicates that viscosity is important to the flow dynamics. The smallest scales are damped out, and only the larger scales remain.

Next time you look at a turbulent flow, try to pick out the smallest and biggest scales of fluid motion. Is the Reynolds number big or small?

Example on the importance of Reynolds number

If an airplane needs testing of its wing, one can make a scaled down model of the wing and test it as a table top model in the lab with the same Reynolds number the actual airplane is subjected to.

For example, if a scale model is one quarter that of the full size, the flow velocity would have to be increased four times. Alternatively, the tests may be conducted in a water tank (water has a higher dynamic viscosity than air), thus maintaining the same Reynolds number.

The results of the laboratory model will be similar to that of the actual plane wing results. Thus we need not bring a full scale plane into the lab and actually test it. This is an example of "dynamic similarity".

Reynolds number is important in the calculation of a body's drag characteristics. A notable example is that of the flow around a cylinder. Above roughly 3×10⁶ Re the drag coefficient drops considerably. This is important when calculating the optimal cruise speeds for low drag (and therefore long range) profiles for airplanes.

Typical values of Reynolds number

Spermatozoa ~ 1×10⁻²
Blood flow in brain ~ 1×10²
Blood flow in aorta ~ 1×10³

Onset of turbulent flow ~ 2.3×10³

Person swimming ~ 4×10⁶
Aircraft ~ 1×10⁷
Blue whale ~ 3×10⁸
A large ship (RMS Queen Elizabeth 2) ~ 5×10⁹

References

Rott, N., “Note on the history of the Reynolds number,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11.

Zagarola, M.V. and Smits, A.J., “Experiments in High Reynolds Number Turbulent Pipe Flow.” AIAApaper #96-0654, 34th AIAA Aerospace Sciences Meeting, Reno, Nevada, January 15 - 18, 1996.

Jermy M., “Fluid Mechanics A Course Reader,” Mechanical Engineering Dept., University of Canterbury, 2005, pp. d5.10.

Hughes, Roger "Civil Engineering Hydraulics," Civil and Environmental Dept., University of Melbourne 1997, pp. 107-152

Fouz, Infaz "Fluid Mechanics," Mechanical Engineering Dept., University of Oxford, 2001, pp96

External links

Gas Dynamics Toolbox Calculate Reynolds number for mixtures of gases using VHS model

Dimensionless numbers | Fluid dynamics | Piping