Reynolds number

In fluid mechanics and heat transfer, the Reynolds number $mathrm\{Re\}$ is a dimensionless number that gives a measure of the ratio of inertial forces () to viscous forces ($mu\; /\; L$) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions.

Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics and heat transfer problems, and as such can be used to determine dynamic similitude between different experimental cases. They are also used to characterise 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 occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce random eddies, vortices and other flow fluctuations.

Reynolds number is named after Osborne Reynolds (1842–1912), who proposed it in 1883. [ cite journal | last = Reynolds | first = Osborne | authorlink = Osborne Reynolds | year = 1883| month = | title = An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels | journal = Philosophical Transactions of the Royal Society | volume = 174 | issue = | pages = 935-982 | id = [http://www.jstor.org/stable/109431 JSTOR] | url = | accessdate = 2008-06-12 | quote = ] [Rott, N., “ [http://dx.doi.org/10.1146/annurev.fl.22.010190.000245 Note on the history of the Reynolds number] ,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11.]

Definition

Reynolds number is defined as [ [http://www.engineeringtoolbox.com/reynolds-number-d_237.html Reynolds Number] (engineeringtoolbox.com)] :

:$mathrm\{Re\}\_L\; =$ ho {old mathrm V} L} over {mu = {old mathrm V} L} over { u

where:
* is the mean fluid velocity in (SI units: m/s)
* $L$ is the "characteristic length" (m)
* $mu$ is the dynamic viscosity of the fluid (Pa·s or N·s/m²)
* $u$ is the "kinematic" viscosity (defined as $u\; =\; mu\; /\; ho$) (m²/s)
* $ho$ is the density of the fluid (kg/m³)

For any shape, the parameter that is used as the characteristic length $L$ is not given explicitly by physics, but rather is chosen by convention (and usually subscripted after the 'Re'). For flow in a pipe for instance, the characteristic length is usually taken as the pipe diameter (so $mathrm\{Re\}\_D$), but is occasionally taken as being the radius (so $mathrm\{Re\}\_r$). So, it is important that for comparison of flows or Reynolds numbers, that it is the same type of characteristic length being employed.

Transition Reynolds number

In boundary layer flow over a flat plate, experiments can confirm that, after a certain length of flow, a laminar boundary layer will become unstable and become turbulent. This instability occurs across different scales and with different fluids, usually when $mathrm\{Re\}\_x\; approx\; 5\; imes\; 10^5$, where $x$ is the distance from the leading edge of the flat plate, and the flow velocity is the 'free stream' velocity of the fluid outside the boundary layer.

For flow in a pipe of diameter $D$, experimental observations show that for 'fully developed' flow, laminar flow occurs when and turbulent flow occurs when $mathrm\{Re\}\_D\; >\; 4000$ [J.P Holman "Heat transfer", McGraw-Hill, 2002, p.207] . In the interval between 2000 and 4000, laminar and turbulent flows are possible ('transition' flows), depending on other factors, such as pipe roughness and flow uniformity). This result is generalised to non-circular channels using the hydraulic diameter, allowing a transition Reynolds number to be calculated for other shapes of channel.

These transition Reynolds numbers are also called "critical Reynolds numbers", and were studied by Osborne Reynolds around 1895 [see Rott] .

Reynolds number in pipe friction

Pressure drops seen for fully-developed flow of fluids through pipes can be predicted using the Moody diagram which is plots the the friction factor $f$ against Reynolds number $\{mathrm\; Re\}\_D$ and relative roughness $epsilon\; /\; D$. The diagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases.

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:

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

: $mathit\{Eu\}^\{star\}\; =\; mathit\{Eu\}\; ;\; quadquad\; mbox\{i.e.\}\; quad\; \{p^\{star\}over\; ho^\{star\}\; \{v^\{star^\{2\; =\; \{pover\; ho\; v^\{2\; ;\; ,$

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.

Typical values of Reynolds number

"Note:" these values are meaningless without a definition of the characteristic length in each case.

* Spermatozoa ~ 1×10⁻²
* Blood flow in brain ~ 1×10²
* Blood flow in aorta ~ 1×10³Onset of turbulent flow ~ 2.3×10³-5.0×10⁴ for pipe flow to 10⁶ for boundary layers
* 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

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 the Reynolds number increases, smaller and smaller scales of the flow are visible. In a 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 (provided the compressibility effects of air are not significant). 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 one thirteenth the size 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.

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.

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.

Reynolds number in viscous fluids

Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The Reynolds number is very small and Stokes Law can be used to measure the viscosity of the fluid. Spheres are allowed to fall through the fluid and they reach the terminal velocity quickly, from which the viscosity can be determined.

The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who exude viscous solutions from their skin to aid flow over their bodies while swimming. It has been used in yacht racing by owners who want to gain a speed advantage by pumping a polymer solution such as low molecular weight polyoxyethylene in water, over the wetted surface of the hull. It is however, a problem for mixing of polymers, because turbulence is needed to distribute fine filler (for example) through the material. Inventions such as the "cavity transfer mixer" have been developed to produce multiple folds into a moving melt so as to improve mixing efficiency. The device can be fitted onto extruders to aid mixing.

ee also

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

References and notes

Further reading

*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) [http://jilawww.colorado.edu/perkinsgroup/Purcell_life_at_low_reynolds_number.pdf]
*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

* [http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate Reynolds number for mixtures of gases using VHS model
* [http://www.calctool.org/CALC/eng/fluid/reynolds Browser-based Reynolds number calculator.]
* [http://www.hrs-spiratube.com/en/resources/comparison-of-laminar-and-turbulent-flow.aspx Practical relation between Reynolds Number and Turbulence]