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. In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units; it does not change if one alters ones system of units of measurement, for example from English units to metric units. ... Fluid mechanics or fluid dynamics is the study of the macroscopic physical behaviour of fluids . ... A full scale X-43 Wind tunnel test. ...

It is named after Osborne Reynolds (1842-1912), who proposed it in 1883. Typically it is given as follows: Osborne Reynolds Osborne Reynolds (23 August 1842 – 21 February 1912) was an Irish engineer. ... 1842 was a common year starting on Saturday (see link for calendar). ... 1912 is a leap year starting on Monday. ... 1883 was a common year starting on Monday (see link for calendar). ...

or

With:

• 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.

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. 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. A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... The Pitch Drop Experiment at the University of Queensland. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Laminar flow (bottom of pic) and turbulent flow (top of pic) over a submarine hull. ... Turbulent flow around an obstacle; the flow further away is laminar Laminar and turbulent water flow over the hull of a submarine Turbulence creating a vortex on an airplane wing In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by low-momentum diffusion, high momentum convection, and...

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 and equal Reynolds numbers. When comparing fluid behaviour at homologous points in a model and a full-scale flow, the following holds:

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. A water channel is an experimental tank for studying resistance and propulsion behaviour of ships, submarines, or other sea vessels. ... A wind tunnel is a research tool developed to assist with studying the effects of air moving over or around solid objects. ... In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units; it does not change if one alters ones system of units of measurement, for example from English units to metric units. ... Mach number (Ma) (pronounced as mack in International English or mock in the American English) is defined as a ratio of speed to the speed of sound in the medium in case. ... A compressible flow is a situation in which the compressibility of the fluid must be taken into account. ... In fluid dynamics, the Froude number (named after William Froude) is the reciprocal of the square root of the Richardson number. ...

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. In fluid dynamics, an eddy is the swirling of a fluid and the reverse current (water) created when the fluid flows past an obstacle. ...

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. The Pitch Drop Experiment at the University of Queensland. ...

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 aeroplane needs testing of its wing, one can make a scaled down small model of the wing and test the wing as a 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.

Related topics

• Hagen-Poiseuille law
• Darcy-Weisbach equation
• Navier-Stokes equations

The Poiseuilles law (or the Hagen-Poiseuille law also named after Gotthilf Heinrich Ludwig Hagen (1797-1884) for his experiments in 1839) is the physical law concerning the voluminal laminar stationary flow ΦV of incompressible uniform viscous liquid (so called Newtonian fluid) through a cylindrical tube with the constant... The Darcy-Weisbach equation is an important and widely used equation in hydraulics. ... In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. ...

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