Reynolds number

In fluid mechanics, the Reynolds number $\mathrm{Re}$ is a dimensionless number that gives a measure of the ratio of inertial forces $\left( {{\rho {\bold \mathrm V}^2} {\bold \mathrm L}^2} \right)$ to viscous forces $\left( {{\mu {\bold \mathrm V}} {\bold \mathrm L}} \right)$ and consequently quantifies the relative importance of these two types of forces for given flow conditions. The concept was introduced by George Gabriel Stokes in 1851,^[1] but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883.^[2]^[3]

Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics problems, and as such can be used to determine dynamic similitude between different experimental cases. They are also used to characterize 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 instabilities.

Definition

Reynolds number can be defined for a number of different situations where a fluid is in relative motion to a surface (the definition of the Reynolds number is not to be confused with the Reynolds Equation or lubrication equation). These definitions generally include the fluid properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension. This dimension is a matter of convention - for example a radius or diameter are equally valid for spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today. Other shapes (such as rectangular pipes or non-spherical objects) have an equivalent diameter defined. For fluids of variable density (e.g. compressible gases) or variable viscosity (non-Newtonian fluids) special rules apply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels.

$\mathrm{Re} = {{\rho {\bold \mathrm V} L} \over {\mu}} = {{{\bold \mathrm V} L} \over {\nu}} = {{{\bold \mathrm Q} L} \over {\nu}A}$ ^[4]

where:

${\bold \mathrm V}$ is the mean fluid velocity (SI units: m/s)
${L}$ is a characteristic linear dimension, (traveled length of fluid, or hydraulic radius when dealing with river systems) (m)
${\mu}$ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s))
${\nu}$ is the kinematic viscosity ( $\nu = \mu /$ ρ) (m²/s)
${\rho}\,$ is the density of the fluid (kg/m³)

And in pipes:

${\mathrm Q}$ is the volumetric flow rate (m³/s)
${A}$ is the pipe cross-sectional area (m²).

Note that this is equal to the ratio between ${\rho {\bold \mathrm V}^2} \over {L}$ , which is the drag (up to a numerical factor, half the drag coefficient), and ${{\mu {\bold \mathrm V}} \over {L}^2}$ , which is the force due to viscosity (up to a numerical factor depending on the form of the flow).

Flow in Pipe

For flow in a pipe or tube, the Reynolds number is generally defined as:^[5]

$\mathrm{Re} = {{\rho {\bold \mathrm V} D_H} \over {\mu}} = {{{\bold \mathrm V} D_H} \over {\nu}} = {{{\bold \mathrm Q} D_H} \over {\nu}A}$

where:

${D_H}$ is the hydraulic diameter of the pipe (m).

Flow in a non-circular duct

For shapes such as squares, rectangular or annular ducts (where the height and width are comparable) the characteristic dimension for internal flow situations is taken to be the hydraulic diameter, $D_H$ , defined as 4 times the cross-sectional area, divided by the wetted perimeter. The wetted perimeter for a channel is the total perimeter of all channel walls that are in contact with the flow.^[6]

$D_H = \frac{4 A}{P}.$

For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter, as can be shown mathematically.

For an annular duct, such as the outer channel in a tube-in-tube heat exchanger, the hydraulic diameter can be shown algebraically to reduce to

$D_{H,\mathrm{annulus}} = D_o - D_i$

where

$D_o$ is the inside diameter of the outside pipe, and

$D_i$ is the outside diameter of the inside pipe.

For calculations involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy.

Flow in a Wide Duct

For a fluid moving between two plane parallel surfaces (where the width is much greater than the space between the plates) then the characteristic dimension is twice the distance between the plates.^[7]

Flow in an Open Channel

For flow of liquid with a free surface, the hydraulic radius must be determined. This is the cross-sectional area of the channel divided by the wetted perimeter. For a semi-circular channel, it is half the radius. For a rectangular channel, the hydraulic radius is the cross-sectional area divided by the wetted perimeter. Some texts then use a characteristic dimension that is 4 times the hydraulic radius (chosen because it gives the same value of Re for the onset of turbulence as in pipe flow),^[8] while others the hydraulic radius as the characteristic length-scale with consequently different values of Re for transition and turbulent flow.

Object in a fluid

The Reynolds number for an object in a fluid, called the particle Reynolds number and often denoted Re_p, is important when considering the nature of flow around that grain, whether or not vortex shedding will occur, and its fall velocity.

Sphere in a fluid

For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid some distance away from the sphere (such that the motion of the sphere does not disturb that reference parcel of fluid). The density and viscosity are those belonging to the fluid.^[9] Note that purely laminar flow only exists up to Re = 0.1 under this definition.

Under the condition of low Re, the relationship between force and speed of motion is given by Stokes' law.^[10]

Oblong object in a fluid

The equation for an oblong object is identical to that of a sphere, with the object being approximated as an ellipsoid and the axis of intermediate length being chosen as the characteristic length scale. Such considerations are important in natural streams, for example, where there are few perfectly spherical grains. For grains in which measurement of each axis is impractical (e.g., because they are too small), sieve diameters are used instead as the characteristic particle length-scale. Both approximations alter the values of the critical Reynolds number.

Fall velocity

The particle Reynolds number is important in determining the fall velocity of a particle. When the particle Reynolds number indicates laminar flow, Stokes' law can be used to calculate its fall velocity. When the particle Reynolds number indicates turbulent flow, a turbulent drag law must be constructed to model the appropriate settling velocity.

Packed Bed

For flow of fluid through a bed of approximately spherical particles of diameter D in contact, if the voidage (fraction of the bed not filled with particles) is ε and the superficial velocity V (i.e. the velocity through the bed as if the particles were not there - the actual velocity will be higher) then a Reynolds number can be defined as:

$\mathrm{Re} = {{\rho {\bold \mathrm V} D} \over {\mu (1-\epsilon)}}.$

Laminar conditions apply up to Re = 10, fully turbulent from 2000.^[9]

Stirred Vessel

In a cylindrical vessel stirred by a central rotating paddle, turbine or propellor, the characteristic dimension is the diameter of the agitator $D$ . The velocity is $N D$ where $N$ is the rotational speed (revolutions per second). Then the Reynolds number is:

$\mathrm{Re} = {{\rho N D^2} \over {\mu}}.$

The system is fully turbulent for values of Re above 10 000.^[11]

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 $\mathrm{Re}_x \approx 5 \times 10^5$ , where $x$ is the distance from the leading edge of the flat plate, and the flow velocity is the freestream velocity of the fluid outside the boundary layer.

For flow in a pipe of diameter $D$ , experimental observations show that for 'fully developed' flow (Note:^[12]), laminar flow occurs when $\mathrm{Re}_D < 2300$ and turbulent flow occurs when $\mathrm{Re}_D > 4000$ ^[13]. In the interval between 2300 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 plots the Darcy–Weisbach friction factor $f$ against Reynolds number ${\mathrm{Re}}$ and relative roughness $\epsilon / D$ . The diagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases. The nature of pipe flow is strongly dependent on whether the flow is laminar or turbulant

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 corresponding points in a model and a full-scale flow, the following holds:

$\mathrm{Re}_m = \mathrm{Re} \;$

$\mathrm{Eu}_m = \mathrm{Eu} \; \quad\quad \mbox{i.e.} \quad {p_m \over \varrho_m {v_m}^{2}} = {p\over \varrho v^{2}} \; ,$

quantities marked with 'm' concern the flow around the model and the others the actual flow. This allows engineers to perform experiments with reduced models in water channels or wind tunnels, and correlate the data to the actual flows, saving on costs during experimentation and on lab time. Note that true dynamic similitude may require matching other dimensionless numbers as well, such as the Mach number used in compressible flows, or the Froude number that governs open-channel 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 modeling to be useful, it requires a fair amount of experience and judgement of the engineer.

Typical values of Reynolds number ^[14]^[15]

Bacteria ~ 1 x 10⁻⁵
Spermatozoa ~ 1 x 10⁻⁴^[16]
Ciliate ~ 1 x 10⁻¹
Smallest Fish ~ 1

Blood flow in brain ~ 1 × 10²
Blood flow in aorta ~ 1 × 10³

Onset of turbulent flow ~ 2.3 × 10³ to 5.0 × 10⁴ for pipe flow to 10⁶ for boundary layers

Typical pitch in Major League Baseball ~ 2 × 10⁵
Person swimming ~ 4 × 10⁶
Fastest Fish ~ 1 x 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 (sometimes 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 of the model would have to be multiplied by a factor of 4 to obtain similar flow behavior.

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 [1]. 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 definition of the Reynolds number 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 a sign of such a condition.

Reynolds number in viscous fluids

Creeping flow past a sphere: streamlines, drag force F_d and force by gravity F_g.

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.

Where does it come from?

The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier-Stokes equations:

$\varrho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}.$

Each term in the above equation has the units of a volume force or, equivalently, an acceleration times a density. Each term is thus dependent on the exact measurements of a flow. When one renders the equation nondimensional, that is that we multiply it by a factor with inverse units of the base equation, we obtain a form which does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the following factor:

$\frac{D}{\varrho V^2}$

where the symbols are the same as those used in the definition of the Reynolds number. If we now set:

$\mathbf{v'} = \frac{\mathbf{v}}{V},\ p' = p\frac{1}{\varrho V^2}, \ \mathbf{f'} = \mathbf{f}\frac{D}{\varrho V^2}, \ \frac{\partial}{\partial t'} = \frac{D}{V} \frac{\partial}{\partial t}, \ \nabla' = D \nabla$

we can rewrite the Navier-Stokes equation without dimensions:

$\frac{\partial \mathbf{v'}}{\partial t'} + \mathbf{v'} \cdot \nabla' \mathbf{v'} = -\nabla' p' + \frac{\mu}{\varrho D V} \nabla'^2 \mathbf{v'} + \mathbf{f'}$

where the term : $\frac{\mu}{\varrho D V} = \frac{1}{\mathrm{Re}}.$

Finally, dropping the primes for ease of reading:

$\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} = -\nabla p + \frac{1}{\mathrm{Re}} \nabla^2 \mathbf{v} + \mathbf{f}.$

This is why mathematically all flows with the same Reynolds number are comparable.

References and notes

↑ Stokes, George (1851). "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums". Transactions of the Cambridge Philosophical Society 9: 8–106.
↑ Reynolds, Osborne (1883). "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". Philosophical Transactions of the Royal Society 174: 935–982. doi:10.1098/rstl.1883.0029. JSTOR.
↑ Rott, N., “Note on the history of the Reynolds number,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 1–11.
↑ www.grc.nasa.gov
↑ Reynolds Number (engineeringtoolbox.com)
↑ J.P. Holman, Heat Transfer, McGraw Hill.
↑ R. W. Fox, A. T. McDonald, Phillip J. Pritchard Introduction to Fluid Mechanics, 6th ed (John Wiley and Sons) ISBN 0 471 20231 2 page 348
↑ V. L. Streeter (1962)Fluid Mechanics, 3rd edn (McGraw-Hill)
↑ ^9.0 ^9.1 M. Rhodes (1989) Introduction to Particle Technology Wiley ISBN 0-471-98482-5 at Google Books
↑ Dusenbery, David B. (2009). Living at Micro Scale, p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
↑ R. K. Sinnott Coulson & Richardson's Chemical Engineering, Volume 6: Chemical Engineering Design, 4th ed (Butterworth-Heinemann) ISBN 0 7506 6538 6 page 473
↑ Full development of the flow occurs as the flow enters the pipe, the boundary layer thickens and then stabilises after several diameters distance into the pipe.
↑ J.P Holman Heat transfer, McGraw-Hill, 2002, p.207
↑ Patel, V. C., W. Rodi, and G. Scheuerer. "Turbulence Models for Near-Wall and Low Reynolds Number Flows- A Review." AIAA Journal 23.9 (1985): 1308-19.
↑ Dusenbery, David B. (2009). Living at Micro Scale, p.136. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
↑ Wiggins, C. H., and R. E. Goldstein. "Flexive and Propulsive Dynamics of Elastica at Low Reynolds Number." Physical Review Letters 80.17 (1998): 3879-82.

External links

Gas Dynamics Toolbox Calculate Reynolds number for mixtures of gases using VHS model
Browser-based Reynolds number calculator.

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