In fluid mechanics, the Reynolds number is a dimensionless quantity that is used to help
predict similar flow patterns in different fluid flow situations. The concept was introduced
by George Gabriel Stokes in 1851, but the Reynolds number is named after Osborne Reynolds,
who popularized its use in 1883. The Reynolds number is defined as the ratio
of inertial forces to viscous forces and consequently quantifies the relative importance of these
two types of forces for given flow conditions. Reynolds numbers frequently arise when performing
scaling of fluid dynamics problems, and as such can be used to determine dynamic similitude
between two different cases of fluid flow. They are also used to characterize different
flow regimes within a similar fluid, 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; turbulent flow occurs at high Reynolds numbers and is
dominated by inertial forces, which tend to produce chaotic eddies, vortices and other
flow instabilities. In practice, matching the Reynolds number
is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic,
and very small changes to shape and surface roughness can result in very different flows.
Nevertheless, Reynolds numbers are a very important guide and are widely used.
Definition The Reynolds number can be defined for several
different situations where a fluid is in relative motion to a surface. 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
radius and diameter are equally valid to describe 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 such as compressible gases or fluids of variable viscosity such as non-Newtonian
fluids, special rules apply. The velocity may also be a matter of convention in some
circumstances, notably stirred vessels. The Reynolds number is defined below for each
case. Flow in pipe
For flow in a pipe or tube, the Reynolds number is generally defined as:
where: is the hydraulic diameter of the pipe; its
characteristic travelled length, ,. is the volumetric flow rate.
is the pipe cross-sectional area. is the mean velocity of the fluid.
is the dynamic viscosity of the fluid). is the kinematic viscosity.
is the density of the fluid. For shapes such as squares, rectangular or
annular ducts where the height and width are comparable, the characteristical dimension
for internal flow situations is taken to be the hydraulic diameter, , defined as:
where A is the cross-sectional area and P is the wetted perimeter. The wetted perimeter
for a channel is the total perimeter of all channel walls that are in contact with the
flow. This means the length of the channel exposed to air is not included in the wetted
perimeter. For a circular pipe, the hydraulic diameter
is exactly equal to the inside pipe diameter, . That is,
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
where is the inside diameter of the outside pipe,
and 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.
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 four times the hydraulic radius, chosen
because it gives the same value of Re for the onset of turbulence as in pipe flow, while
others use the hydraulic radius as the characteristic length-scale with consequently different values
of Re for transition and turbulent flow. Flow around airfoils
Reynolds numbers are used in airfoil design to manage "Scale Effect" when computing/comparing
characteristics. Fluid dynamicists define the chord Reynolds number, R, like this: R
= Vc / ν where V is the flight speed, c is the chord, and ν is the kinematic viscosity
of the fluid in which the airfoil operates, which is 1.460x10−5 m2/s for the atmosphere
at sea level. In some special studies a characteristic length other than chord may be used; rare
is the "span Reynolds number" which is not to be confused with span-wise stations on
a wing where chord is still used. Object in a fluid
The Reynolds number for an object in a fluid, called the particle Reynolds number and often
denoted Rep, is important when considering the nature of the surrounding flow, whether
or not vortex shedding will occur, and its fall velocity.
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 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. 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. Note that
purely laminar flow only exists up to Re = 10 under this definition.
Under the condition of low Re, the relationship between force and speed of motion is given
by Stokes' law. 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 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,
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 fluid flow through a bed of approximately spherical particles of diameter D in contact,
if the "voidage" is ε and the "superficial velocity" is vs, the Reynolds number can be
defined as:
or
The choice of equation depends on the system involved: the first is successful in correlating
the data for various types of packed and fludized beds, the second Reynolds number suits for
the liquid-phase data, while the third was found successful in correlating the fludized
bed data, being first introduced for liquid fluidized bed system.
Laminar conditions apply up to Re = 10, fully turbulent from 2000.
Stirred vessel In a cylindrical vessel stirred by a central
rotating paddle, turbine or propeller, the characteristic dimension is the diameter of
the agitator . The velocity is where is the rotational speed. Then the Reynolds number
is:
The system is fully turbulent for values of Re above 10 000.
Transition and turbulent flow In boundary layer flow over a flat plate,
experiments confirm that, after a certain length of flow, a laminar boundary layer will
become unstable and turbulent. This instability occurs across different scales and with different
fluids, usually when , where 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 , experimental observations show that for "fully developed"
flow, laminar flow occurs when and turbulent flow occurs when . In the interval between
2300 and 4000, laminar and turbulent flows are possible and are called "transition" flows,
depending on other factors, such as pipe roughness and flow uniformity. This result is generalized
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. The
critical Reynolds number is different for every geometry.
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 against Reynolds number and
relative roughness . 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 turbulent. 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 behavior at corresponding points in
a model and a full-scale flow, the following holds:
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, 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. 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 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.
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.
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, a bulge of atheroma
may be the cause of turbulent flow, where audible turbulence may be detected with a
stethoscope. Derivation
The Reynolds number can be obtained when one uses the nondimensional form of the incompressible
Navier-Stokes equations for a newtonian fluid expressed in the lagrangian derivative:
Each term in the above equation has the units of a "body force" with the same dimensions
of a density times an acceleration. Each term is thus dependent on the exact measurements
of a flow. When one renders the equation nondimensional, that is when 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:
where: is the mean velocity, or , relative to the
fluid. is the characteristic length, ,.
is the fluid density. If we now set:
we can rewrite the Navier-Stokes equation without dimensions:
where the term Finally, dropping the primes for ease of reading:
This is why mathematically all newtonian, incompressible flows with the same Reynolds
number are comparable. Notice also, in the above equation, as the viscous terms vanish.
Thus, high Reynolds number flows are approximately inviscid in the free-stream.
See also Reynolds transport theorem
Drag coefficient Deposition
Notes
References
Further reading Zagarola, M. V. and Smits, A. J., "Experiments
in High Reynolds Number Turbulent Pipe Flow." AIAA paper #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, p. 96
E. M. Purcell. "Life at Low Reynolds Number", American Journal of Physics vol 45, pp. 3–11[1]
Truskey, G. A., Yuan, F, Katz, D. F.. Transport Phenomena in Biological Systems Prentice Hall,
pp. 7. ISBN 0-13-042204-5. ISBN 978-0-13-042204-0. External links
The Reynolds Number at Sixty Symbols