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The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place.
around a cylinder. This can occur around cylinders and spheres, for any fluid, cylinder size and fluid speed, provided that the flow has a Reynolds number between roughly 40 and 1000.
, introduced Reynolds numbers
popularised the concept
The Reynolds number (Re) is an important
used to help predict flow patterns in different fluid flow situations. It has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. The Reynolds Number is used to predict the transition from
of flow, and used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.
The concept was introduced by
in 1851, but the Reynolds number was named by
in 1908 after
(), who popularized its use in 1883.
The Reynolds number is the
of inertial forces to
forces within a fluid which is subjected to relative internal movement due to different fluid velocities, in what is known as a
in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.
This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine
between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not linear and the application of Reynolds numbers to both situations allows scaling factors to be developed.
With respect to
laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, c
turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic ,
and other flow instabilities.
The Reynolds number is defined as
{\displaystyle \mathrm {Re} ={\frac {\rho vL}{\mu }}={\frac {vL}{\nu }}}
of the fluid (: kg/m3)
v is a characteristic velocity of the fluid with respect to the object (m/s)
L is a characteristic linear dimension (m)
(Pa·s or N·s/m2 or kg/(m·s))
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
or characteristic dimension (L in the above equation). 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 , special rules apply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels.
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.
Osborne Reynolds' apparatus of 1883 demonstrating the onset of turbulent flow. The apparatus is still at the University of Manchester.
diagram from Reynolds' 1883 paper showing onset of turbulent flow.
Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes
to . In his 1883 paper Reynolds described the transition from laminar to turbulent flow in a classic experiment in which he examined the behaviour of water flow under different flow velocities using a small stream of dyed water introduced into the centre of clear water flow in a larger pipe.
The larger pipe was glass so the behaviour of the layer of the dyed stream could be observed, and at the end of this pipe there was a flow control valve used to vary the water velocity inside the tube. When the velocity was low, the dyed layer remained distinct through the entire length of the large tube. When the velocity was increased, the layer broke up at a given point and diffused throughout the fluid's cross-section. The point at which this happened was the transition point from laminar to turbulent flow.
From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of
forces. Reynolds also proposed what is now known as the
of turbulent flows, where quantities such as
are expressed as the sum of mean and fluctuating components. Such averaging allows for 'bulk' description of turbulent flow, for example using the .
The form of the Reynolds number can be derived as follows
inertial forces
viscous forces
acceleration
dynamic viscosity
{\displaystyle {\begin{aligned}\mathrm {Re} &={\frac {\mbox{inertial forces}}{\mbox{viscous forces}}}={\frac {({\mbox{mass}})({\mbox{acceleration}})}{({\mbox{dynamic viscosity}})\left({\frac {\mbox{velocity}}{\mbox{distance}}}\right)({\mbox{area}})}}\\&={\frac {(\rho L^{3})\left({\frac {v}{t}}\right)}{\mu \left({\frac {v}{L}}\right)L^{2}}}={\frac {(\rho L^{3})\left({\frac {1}{t}}\right)}{\mu \left({\frac {1}{L}}\right)L^{2}}}={\frac {(\rho L^{2})\left({\frac {1}{t}}\right)}{\mu }}={\frac {(\rho )\left({\frac {L}{t}}\right)(L)}{\mu }}={\frac {\rho vL}{\mu }}={\frac {vL}{\nu }}\end{aligned}}}
v is the maximum velocity of the object relative to the fluid (: m/s)
L is a characteristic linear dimension, (travelled
when dealing with river systems) (m)
t denotes the
(Pa·s or N·s/m2 or kg/(m.s))
(nu) is the
(ν = μ/ρ) (m2/s)
of the fluid (kg/m3).
Note that multiplying the Reynolds number by Lv/Lv yields ρv2L2/μvL, which is the ratio of the inertial forces to the viscous forces. It could also be considered the ratio of the total momentum transfer to the molecular momentum transfer.
or tube, the Reynolds number is generally defined as:
{\displaystyle \mathrm {Re} ={\frac {\rho {v}D_{\mathrm {H} }}{\mu }}={\frac {{v}D_{\mathrm {H} }}{\nu }}={\frac {{Q}D_{\mathrm {H} }}{\nu A}}}
of the pipe (the inside diameter if the pipe is circular) (m).
Q is the volumetric
A is the pipe's cross-sectional area (m2).
υ is the mean velocity of the fluid (m/s).
(Pa·s = N·s/m2 = kg/(m*s)).
ν (nu) is the
(ν = μ/ρ) (m2/s).
of the fluid (kg/m3).
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 , DH, defined as:
{\displaystyle D_{\mathrm {H} }={\frac {4A}{P}},}
where A is the cross-sectional area and P is the . 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, D. That is,
{\displaystyle D_{\mathrm {H} }=D.}
For an annular duct, such as the outer channel in a tube-in-tube , the hydraulic diameter can be shown algebraically to reduce to
{\displaystyle D_{\mathrm {H,annulus} }=D_{\mathrm {o} }-D_{\mathrm {i} }}
Do is the inside diameter of the outside pipe, and
Di 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, if the aspect ratio AR of the duct cross-section remains in the range 1/4 & AR & 4.
Main article:
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 Rex ≈ 5×105, where x is the distance from the leading edge of the flat plate, and the flow velocity is the
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 ReD & 2000 and turbulent flow occurs when ReD & 4000. In the interval between 2000 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 , allowing a transition Reynolds number to be calculated for other shapes of channel.
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.
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 equal to the distance between the plates.
For flow of liquid with a free surface, the
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.
Reynolds numbers are used in
design to (among other things) manage "Scale Effect" when computing/comparing characteristics (a tiny wing, scaled to be huge, will perform differently). Fluid dynamicists define the chord Reynolds number, R, like this: R = Vc/ν where V is the flight speed, c is the chord length, and ν is the kinematic viscosity of the fluid in which the airfoil operates, which is 1.460×10-5 m2/s for the atmosphere at . In some special studies a characteristic length other th rare is the "span Reynolds number" which is not to be confused with spanwise stations on a wing where chord is still used.
The Reynolds number for an object moving in a fluid, called the particle Reynolds number and often denoted Rep. It characterizes the nature of the surrounding flow and its fall velocity.
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Creeping flow past a falling sphere: , drag force Fd and force by gravity Fg.
Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The Reynolds number is very small and
can be used to measure the
of the fluid. Spheres are allowed to fall through the fluid and they reach the
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
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
efficiency. The device can be fitted onto
to aid mixing.
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 .
The equation for an oblong object is identical to that of a sphere, with the object being approximated as an
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.
The particle Reynolds number is important in determining the fall velocity of a particle. When the particle Reynolds number indicates laminar flow,
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.
For fluid flow through a bed, of approximately spherical particles of diameter D in contact, if the "" is ε and the "" is vs, the Reynolds number can be defined as:
{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu }}.}
{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu \varepsilon }}.}
{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu (1-\varepsilon )}}.}
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 Re = 2000.
In a cylindrical vessel stirred by a central rotating paddle, turbine or propeller, the characteristic dimension is the diameter of the agitator D. The velocity V is ND where N is the . Then the Reynolds number is:
{\displaystyle \mathrm {Re} ={\frac {\rho ND^{2}}{\mu }}={\frac {\rho VD}{\mu }}.}
The system is fully turbulent for values of Re above 10000.
The , which describes the
f as a function of the Reynolds number and relative pipe roughness.
Pressure drops seen for fully developed flow of fluids through pipes can be predicted using the
which plots the
f against Reynolds number Re and relative roughness ε/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 turbulent.
Qualitative behaviors of fluid flow over a cylinder depends to a large extent on R similar flow patterns often appear when the shape and Reynolds number is matched, although other parameters like surface roughness have a big effect
In order for two flows to be similar they must have the same geometry, and have equal Reynolds numbers and . When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds:
{\displaystyle \mathrm {Re} _{\mathrm {m} }=\mathrm {Re} }
{\displaystyle \mathrm {Eu} _{\mathrm {m} }=\mathrm {Eu} }
{\displaystyle \mathrm {Re} _{\mathrm {m} }}
is the Reynolds number for the model and
{\displaystyle \mathrm {Re} }
is Reynolds and similarly for
{\displaystyle \mathrm {Eu} _{\mathrm {m} }}
the Euler number.
The model numbers and design numbers should be in the same proportion, hence:
{\displaystyle {\frac {p_{\mathrm {m} }}{\rho _{\mathrm {m} }{v_{\mathrm {m} }}^{2}}}={\frac {p}{\rho v^{2}}}}
This allows engineers to perform experiments with reduced scale models in
or , 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
as well, such as the
used in , or the
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.
An example where the mere Reynolds number is not sufficient for similarity of flows (or even the flow regime – laminar or turbulent) are bounded flows, i.e. flows that are restricted by walls or other boundaries. A classical example of this is the , where the dimensionless ratio of radii of bounding cylinders is also important, and many technical applications where these distinctions play an important role. Principles of these restrictions were developed by
and developed further by
Typical values of Reynolds number
~ 1 x 10-1
~ 1 × 102
Blood flow in
~ 1 × 103
Onset of turbulent flow ~ 2.3 × 103 to 5.0 × 104 for pipe flow to 106 for boundary layers
Typical pitch in
~ 2 × 105
~ 4 × 106
~ 4 × 108
A large ship () ~ 5 × 109
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 alw 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.
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on blood circulation in the body is dependent on . 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
may be the cause of turbulent flow, where audible turbulence may be detected with a stethoscope.
Reynolds number interpretation has been extended into the area of arbitrary . Such as financial flows, nonlinear networks, etc. In the latter case an artificial viscosity is reduced to nonlinear mechanism of energy distribution in
media. Reynolds number then represents a basic control parameter which expresses a balance between injected and dissipated energy flows for open boundary system. It has been shown
that Reynolds critical regime separates two types of phase space motion: accelerator (attractor) and decelerator. High Reynolds number leads to a chaotic regime transition only in frame of
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The Reynolds number can be obtained when one uses the
form of the incompressible
for a newtonian fluid expressed in terms of the :
{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {f} .}
Each term in the above equation has the units of a "body force" (force per unit volume) 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:
{\displaystyle L \over \rho V^{2}}
V is the mean velocity, v or v, relative to the fluid (m/s).
L is the characteristic length (m).
ρ is the fluid density (kg/m3).
If we now set:
{\displaystyle \mathbf {v'} ={\frac {\mathbf {v} }{V}},\quad p'=p{\frac {1}{\rho V^{2}}},\quad \mathbf {f'} =\mathbf {f} {\frac {L}{V^{2}}},\quad {\frac {\partial }{\partial t'}}={\frac {L}{V}}{\frac {\partial }{\partial t}},\quad \nabla '=L\nabla }
we can rewrite the Navier–Stokes equation without dimensions:
{\displaystyle {\frac {D\mathbf {v'} }{Dt'}}=-\nabla 'p'+{\frac {\mu }{\rho LV}}\nabla '^{2}\mathbf {v'} +\mathbf {f'} }
where the term μ/ρLV = 1/Re.
Finally, dropping the primes for ease of reading:
{\displaystyle {\frac {D\mathbf {v} }{Dt}}=-\nabla p+{\frac {1}{\mathrm {Re} }}\nabla ^{2}\mathbf {v} +\mathbf {f} .}
This is why mathematically all Newtonian, incompressible flows with the same Reynolds number are comparable. Notice also, in the above equation, as Re → ∞ the viscous terms vanish. Thus, high Reynolds number flows are approximately inviscid in the free-stream.
There are many . The Reynolds number measures the ratio of advection and diffusion effects on structures in the velocity field, and is therefore closely related to , which measure the ratio of these effects on other fields carried by the flow, for example temperature and magnetic fields. Replacement of the
ν = μ/ρ in Re by the thermal or magnetic diffusivity results in respectively the thermal Péclet number and the . These are therefore related to Re by products with ratios of diffusivities, namely the
The definition of the Reynolds number is not to be confused with the
or lubrication equation.
Full development of the flow occurs as the flow enters the pipe, the boundary layer thickens and then stabilizes after several diameters distance into the pipe.
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