In fluid mechanics, the Reynolds number may be described as the ratio of inertial forces (v_sρ) to viscous forces (μ/L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions. This box:      Fluid mechanics is the study of how fluids move and the forces on them. ... This article is about the mathematical concept. ... This article is about inertia as it applies to local motion. ... For other uses, see Force (disambiguation). ... For other uses, see Viscosity (disambiguation). ...

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, and will have similar flow geometry. In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... A full scale X-43 Wind tunnel test. ...

It is also used to identify and predict different flow regimes, such as laminar or turbulent flow. 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, which tend to produce random eddies, vortices and other flow fluctuations. laminar and turbulent water flow over the hull of a submarine In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion, low momentum convection, and pressure and velocity independence from time. ... 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... Laminar flow (bottom) and turbulent flow (top) 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...

It is named after Osborne Reynolds (1842–1912), who proposed it in 1883.^[1] Osborne Reynolds Osborne Reynolds (23 August 1842–21 February 1912) was a British fluid dynamics engineer. ... 1842 was a common year starting on Saturday (see link for calendar). ... 1912 (MCMXII) was a leap year starting on Monday in the Gregorian calendar (or a leap year starting on Tuesday in the 13-day-slower Julian calendar). ... Year 1883 (MDCCCLXXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ...

Definition

Typically it is given as follows^[2]:

where:

• v_s is the mean fluid velocity in ms^-1
• L is the characteristic length in m
• μ is the (absolute) dynamic fluid viscosity in Nsm^-2 or Pa·s
• ν is the kinematic fluid viscosity, defined as ν = μ/ρ, in m²s^-1
• ρ is the density of the fluid in kgm^-3

For any shape, the parameter that is chosen to be the characteristic length isn't given by physics, but is a convention. For flow in a pipe for instance, the characteristic length is the pipe diameter in some literature; while being the radius (half of the diameter) in other literature. So, it is important that for comparison of flows or Reynolds numbers, that it is the same type of characteristic length being employed. A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... For other uses, see Viscosity (disambiguation). ... For other uses, see Density (disambiguation). ...

For flow over a flat plate, the characteristic length is usually the length of the plate and the characteristic velocity is the free stream velocity. In a boundary layer over a flat plate the local regime of the flow is determined by the Reynolds number based on the distance measured from the leading edge of the plate. In this case, the transition to turbulent flow occurs at a Reynolds number of the order of 10⁵ or 10⁶.

For non identical geometries, approximate formulas for the characteristic length exist to make Reynolds numbers comparable. An example is the hydraulic diameter, for a non-circular cross section pipe. The hydraulic diameter, , is a commonly used term when handling flow in noncircular tubes and channels. ...

The similarity of flows

In order for two flows to be similar they must have the same geometry, and 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: The Euler number or cavitation number is a dimensionless number used in flow calculations. ...

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. L, the characteristic length(for pipelines it would be the pipeline diameter), can be best calculated by finding the squares of frontal length and width and then square rooting the sum. A water channel is an experimental tank for studying resistance and propulsion behaviour of ships, submarines, or other sea vessels. ... NASA wind tunnel with the model of a plane 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. ... An F/A-18 Hornet breaking the sound barrier. ... A compressible flow is a situation in which the compressibility of a fluid must be taken into account. ... The Froude number is a dimensionless number used to quantify the resistance of an object moving through water, and compare objects of different sizes. ...

The critical Reynolds number

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 many engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 3000 to ensure that the flow is either laminar or turbulent.

Reynolds number sets the smallest scales of turbulent motion

In a turbulent flow, there is a range of scales of the time-varying fluid motion. The size of the largest scales of fluid motion (sometime called 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 velocity perturbations 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. The largest eddies will always be the same size; the smallest eddies are determined by the Reynolds number.

What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important at large scales of the flow. With a strong predominance of inertial forces over viscous forces, the largest scales of fluid motion are undamped -- there is not enough viscosity to dissipate their motions. The kinetic energy must "cascade" from these large scales to progressively smaller scales until a level is reached for which the scale is small enough for viscosity to become important (that is, viscous forces become of the order of inertial ones). It is at these small scales where the dissipation of energy by viscous action finally takes place. The Reynolds number indicates at what scale this viscous dissipation occurs. Therefore, since the largest eddies are dictated by the flow geometry and the smallest scales are dictated by the viscosity, the Reynolds number can be understood as the ratio of the largest scales of the turbulent motion to the smallest scales.

Example of the importance of the Reynolds number

If an airplane wing needs testing, one can make a scaled down model of the wing and test it in a wind tunnel using the same Reynolds number that the actual airplane is subjected to. If for example the scale model has linear dimensions one quarter of full size, the flow velocity would have to be increased four times to obtain similar flow behaviour.

Alternatively, tests could be conducted in a water tank instead of in air. As the kinematic viscosity of water is around 13 times less than that of air at 15 °C, in this case the scale model would need to be about 13 times smaller in all dimensions to maintain the same Reynolds number, assuming the full-scale flow velocity was used.

The results of the laboratory model will be similar to those of the actual plane wing results. Thus there is no need to 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. An object moving through a gas or liquid experiences a force in direction opposite to its motion. ... The drag coefficient (Cd, Cx or Cw, depending on the country) is a dimensionless quantity that describes a characteristic amount of aerodynamic drag caused by fluid flow, used in the drag equation. ...

Reynolds number in physiology

Poiseuille's law on blood circulation in the body is dependent on laminar flow. In turbulent flow the flow rate is proportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressure gradient in laminar flow. 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... Laminar flow (bottom) and turbulent flow (top) over a submarine hull. ...

Using the Reynolds equation we can see that a large diameter, with rapid flow, where the density of the blood is high tends towards turbulence. Rapid changes in vessel diameter may lead to turbulent flow, for instance when a narrower vessel widens to a larger one. Furthermore, an atheroma may be the cause of turbulent flow, and as such detecting turbulence with a stethoscope may be an indication of such a condition. In pathology, an atheroma (plural: atheromata) is an accumulation and swelling (-oma) in artery walls that is made up of cells, or cell debris, that contain lipids (cholesterol and fatty acids), calcium and a variable amount of fibrous connective tissue. ...

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 ~ 2.3×10³-5.0×10⁴ for pipe flow to 10⁶ for boundary layers Schematic diagram of a sperm cell, showing the (1) acrosome, (2) cell membrane, (3) nucleus, (4) mitochondria, and (5) flagellum (tail) A sperm cell, or spermatozoon ( spermatozoa) (in Greek: sperm = semen and zoon = alive), is the haploid cell that is the male gamete. ... Blood flow is the flow of blood in the cardiovascular system. ... Human brain In animals, the brain (enkephale) (Greek for in the skull), is the control center of the central nervous system, responsible for behavior. ... The aorta (generally pronounced [eɪˈɔːtÉ™] or ay-orta) is the largest artery in the human body, originating from the left ventricle of the heart and bringing oxygenated blood to all parts of the body in the systemic circulation. ...

• Typical pitch in Major League Baseball ~ 2×10⁵
• Person swimming ~ 4×10⁶
• Blue Whale ~ 3×10⁸
• A large ship (RMS Queen Elizabeth 2) ~ 5×10⁹

Major Leagues redirects here. ... Swimmer redirects here. ... Binomial name (Linnaeus, 1758) Blue Whale range Subspecies B. m. ... RMS Queen Elizabeth 2 (QE2) is a Cunard Line ocean liner named after the earlier Cunard liner RMS Queen Elizabeth. ...

See also

• Darcy-Weisbach equation
• Hagen-Poiseuille law
• Navier-Stokes equations
• Reynolds transport theorem

The Darcy-Weisbach equation is an important and widely used equation in hydraulics. ... 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 Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... Reynolds Transport Theorem Reynolds transport theorem is a fundamental theorem used in formulating the basic laws of fluid dynamics. ...

References

• [1] 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
• E.M. Purcell. "Life at Low Reynolds Number", American Journal of Physics vol 45, p. 3-11 (1977)[2]
• Truskey, G.A., Yuan, F, Katz, D.F. (2004). Transport Phenomena in Biological Systems Prentice Hall, pp. 7. ISBN-10: 0130422045. ISBN-13: 978-0130422040.

External links

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