In
fluid dynamics,
turbulence or
turbulent flow is a
fluid regime characterized by chaotic,
stochastic property changes. This includes low
momentum diffusion, high momentum
convection, and rapid variation of
pressure and
velocity in space and time. Flow that is not
turbulent is called
laminar flow. While
there is no theorem relating
Reynolds
number to turbulence, flows with high Reynolds numbers usually
become turbulent, while those with low Reynolds numbers usually
remain laminar. For pipe flow, a Reynolds number above about 4000
will most likely correspond to turbulent flow, while a Reynold's
number below 2100 indicates laminar flow. The region in between
(2100 Re 4000) is called the transition region. In turbulent flow,
unsteady vortices appear on many scales and interact with each
other.
Drag due to
boundary layer skin friction increases. The
structure and location of boundary layer separation often changes,
sometimes resulting in a reduction of overall drag. Although
laminarturbulent
transition is not governed by
Reynolds number, the same transition occurs
if the size of the object is gradually increased, or the
viscosity of the fluid is decreased, or if the
density of the fluid is increased.
Turbulence causes the formation of
eddies of many different length
scales. Most of the kinetic energy of the turbulent motion is
contained in the large scale structures. The energy "cascades" from
these large scale structures to smaller scale structures by an
inertial and essentially
inviscid
mechanism. This process continues, creating smaller and smaller
structures which produces a hierarchy of eddies. Eventually this
process creates structures that are small enough that molecular
diffusion becomes important and viscous dissipation of energy
finally takes place. The scale at which this happens is the
Kolmogorov length
scale.
Turbulent diffusion is usually described by a turbulent
diffusion coefficient. This turbulent
diffusion coefficient is defined in a phenomenological sense, by
analogy with the molecular diffusivities, but it does not have a
true physical meaning, being dependent on the flow conditions, and
not a property of the fluid, itself. In addition, the turbulent
diffusivity concept assumes a constitutive relation between a
turbulent flux and the gradient of a mean variable similar to the
relation between flux and gradient that exists for molecular
transport. In the best case, this assumption is only an
approximation. Nevertheless, the turbulent diffusivity is the
simplest approach for quantitative analysis of turbulent flows, and
many models have been postulated to calculate it. For instance, in
large bodies of water like oceans this coefficient can be found
using
Richardson's fourthird
power law and is governed by the
random
walk principle. In rivers and large ocean currents, the
diffusion coefficient is given by variations of Elder's
formula.
When designing piping systems, turbulent flow requires a higher
input of energy from a pump (or fan) than laminar flow. However,
for applications such as heat exchangers and reaction vessels,
turbulent flow is essential for good heat transfer and
mixing.
While it is possible to find some particular solutions of the
NavierStokes equations
governing fluid motion, all such solutions are unstable at large
Reynolds numbers. Sensitive dependence on the initial and boundary
conditions makes fluid flow irregular both in time and in space so
that a statistical description is needed.
Russian
mathematician Andrey Kolmogorov
proposed the first statistical theory of turbulence, based on the
aforementioned notion of the energy cascade (an idea originally
introduced by Richardson) and
the concept of selfsimilarity. As a result, the
Kolmogorov microscales were named
after him. It is now known that the selfsimilarity is broken so
the statistical description is presently modified. Still, the
complete description of turbulence remains one of the
unsolved problems in physics.
According to an apocryphal story
Werner Heisenberg was asked what he would
ask
God, given the opportunity. His reply was:
"When I meet God, I am going to ask him two questions: Why
relativity? And why turbulence? I
really believe he will have an answer for the first." A similar
witticism has been attributed to
Horace
Lamb (who had published a noted text book on
Hydrodynamics)—his choice being
quantum electrodynamics (instead of
relativity) and turbulence. Lamb was quoted as saying in a speech
to the
British
Association for the Advancement of Science, "I am an old man
now, and when I die and go to heaven there are two matters on which
I hope for enlightenment. One is
quantum electrodynamics, and the
other is the turbulent motion of fluids. And about the former I am
rather optimistic."
Examples of turbulence
Laminar and turbulent flow of
cigarette smoke.
 Smoke rising from a cigarette. For the first few centimeters, the flow
remains laminar, and then becomes unstable and turbulent as the
rising hot air accelerates upwards. Similarly, the dispersion of pollutants in
the atmosphere is governed by turbulent processes.
 Flow over a golf ball. (This can be
best understood by considering the golf ball to be stationary, with
air flowing over it.) If the golf ball were smooth, the boundary
layer flow over the front of the sphere would be laminar at typical
conditions. However, the boundary layer would separate early, as
the pressure gradient switched from favorable (pressure decreasing
in the flow direction) to unfavorable (pressure increasing in the
flow direction), creating a large region of low pressure behind the
ball that creates high form drag. To
prevent this from happening, the surface is dimpled to perturb the
boundary layer and promote transition to turbulence. This results
in higher skin friction, but moves the point of boundary layer
separation further along, resulting in lower form drag and lower
overall drag.
 The mixing of warm and cold air in the atmosphere by wind,
which causes clearair
turbulence experienced during airplane flight, as well as poor
astronomical seeing (the
blurring of images seen through the atmosphere.)
 Most of the terrestrial atmospheric circulation
 The oceanic and atmospheric mixed
layers and intense oceanic currents.
 The flow conditions in many industrial equipment (such as
pipes, ducts, precipitators, gas scrubbers,
dynamic scraped
surface heat exchangers, etc.) and machines (for instance,
internal combustion
engines and gas turbines).
 The external flow over all kind of vehicles such as cars,
airplanes, ships and submarines.
 The motions of matter in stellar atmospheres.
 A jet exhausting from a nozzle into a quiescent fluid. As the
flow emerges into this external fluid, shear layers originating at
the lips of the nozzle are created. These layers separate the fast
moving jet from the external fluid, and at a certain critical
Reynolds number they become unstable
and break down to turbulence.
 Race cars unable to follow each other through fast corners due
to turbulence created by the leading car causing understeer.
 In windy conditions, trucks that are on the motorway gets
buffeted by their wake.
 Round bridge supports under water. In the summer when the river
is flowing slowly the water goes smoothly around the support legs.
In the winter the flow is faster, so a higher Reynolds Number, so
the flow may start off laminar but is quickly separated from the
leg and becomes turbulent.
Kolmogorov's Theory of 1941
Richardson's notion of turbulence was that a turbulent flow is
composed by "eddies" of different sizes. The sizes define a
characteristic length scale for the eddies, which are also
characterized by velocity scales and time scales (turnover time)
dependent on the length scale. The large eddies are unstable and
eventually break up originating smaller eddies, and the kinetic
energy of the initial large eddy is divided into the smaller eddies
that stemmed from it. These smaller eddies undergo the same
process, giving rise to even smaller eddies which inherit the
energy of their predecessor eddy, and so on. In this way, the
energy is passed down from the large scales of the motion to
smaller scales until reaching a sufficiently small length scale
such that the viscosity of the fluid can effectively dissipate the
kinetic energy into internal energy.
In his original theory of 1941,
Kolmogorov postulated that for very high
Reynolds number, the small scale turbulent
motions are statistically isotropic (i.e. no preferential spatial
direction could be discerned). In general, the large scales of a
flow are not isotropic, since they are determined by the particular
geometrical features of the boundaries (the size characterizing the
large scales will be denoted as
L). Kolmogorov's idea was
that in the Richardson's energy cascade this geometrical and
directional information is lost, while the scale is reduced, so
that the statistics of the small scales has a universal character:
they are the same for all turbulent flows when the Reynolds number
is sufficiently high.
Thus, Kolmogorov introduced a second hypothesis: for very high
Reynolds numbers the statistics of small scales are universally and
uniquely determined by the viscosity (\nu) and the rate of energy
dissipation (\varepsilon). With only these two parameters, the
unique length that can be formed by dimensional analysis is
 \eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}.
This is today known as the Kolmogorov length scale (see
Kolmogorov microscales).
A turbulent flow is characterized by a hierarchy of scales through
which the energy cascade takes place. Dissipation of kinetic energy
takes place at scales of the order of Kolmogorov length \eta, while
the input of energy into the cascade comes from the decay of the
large scales, of order
L. These two scales at the extremes
of the cascade can differ by several orders of magnitude at high
Reynolds numbers. In between there is a range of scales (each one
with its own characteristic length
r) that has formed at
the expense of the energy of the large ones. These scales are very
large compared with the Kolmogorov length, but still very small
compared with the large scale of the flow (i.e. \eta \ll r \ll L).
Since eddies in this range are much larger than the dissipative
eddies that exist at Kolmogorov scales, kinetic energy is
essentially not dissipated in this range, and it is merely
transferred to smaller scales until viscous effects become
important as the order of the Kolmogorov scale is approached.
Within this range inertial effects are still much larger than
viscous effects, and it is possible to assume that viscosity does
not play a role in their internal dynamics (for this reason this
range is called "inertial range").
Hence, a third hypothesis of Kolmogorov was that at very high
Reynolds number the statistics of scales in the range \eta \ll r
\ll L are universally and uniquely determined by the scale
r and the rate of energy dissipation \varepsilon.
The way in which the kinetic energy is distributed over the
multiplicity of scales is a fundamental characterization of a
turbulent flow. For homogeneous turbulence (i.e., statistically
invariant under translations of the reference frame) this is
usually done by means of the
energy spectrum function
E(k), where
k is the modulus of the wavevector
corresponding to some harmonics in a Fourier representation of the
flow velocity field
u(
x):
 \mathbf{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3}
\widehat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}}
\mathrm{d}^3\mathbf{k},
where
û(
k) is the Fourier
transform of the velocity field. Thus,
E(
k)d
k represents the contribution to
the kinetic energy from all the Fourier modes with
k

k
k + d
k, and therefore,
 \mathrm{Total\,\, kinetic\,\, energy} =
\int_{0}^{\infty}E(k)\mathrm{d}k.
The wavenumber
k corresponding to length scale
r
is k=2\pi/r. Therefore, by dimensional analysis, the only possible
form for the energy spectrum function according with the third
Kolmogorov's hypothesis is
 E(k) = C \varepsilon^{2/3} k^{5/3} ,
where
C would be a universal constant. This is one of the
most famous results of Kolmogorov 1941 theory, and considerable
experimental evidence has accumulated that supports it.
In spite of this success, Kolmogorov theory is at present under
revision. This theory implicitly assumes that the turbulence is
statistically selfsimilar at different scales. This essentially
means that the statistics are scaleinvariant in the inertial
range. A usual way of studying turbulent velocity fields is by
means of velocity increments:
 \delta \mathbf{u}(r) = \mathbf{u}(\mathbf{x} + \mathbf{r}) 
\mathbf{u}(\mathbf{x});
that is, the difference in velocity between points separated by a
vector
r (since the turbulence is assumed
isotropic, the velocity increment depends only on the modulus of
r).Velocity increments are useful because they
emphasize the effects of scales of the order of the separation
r when statistics are computed. The statistical
scaleinvariance implies that the scaling of velocity increments
should occur with a unique scaling exponent \beta, so that when
r is scaled by a factor \lambda,
 \delta \mathbf{u}(\lambda r)
should have the same statistical distribution as
 \lambda^{\beta}\delta \mathbf{u}(r),
with \beta independent of the scale
r. From this fact, and
other results of Kolmogorov 1941 theory, it follows that the
statistical moments of the velocity increments (known as
structure functions in turbulence) should scale as
 \langle [\delta \mathbf{u}(r)]^n \rangle = C_n
\varepsilon^{n/3} r^{n/3},
where the brackets denote the statistical average, and the C_n
would be universal constants.
There is considerable evidence that turbulent flows deviate from
this behavior. The scaling exponents deviate from the
n/3
value predicted by the theory, becoming a nonlinear function of
the order
n of the structure function. The universality of
the constants have also been questioned. For low orders the
discrepancy with the Kolmogorov
n/3 value is very small,
which explain the success of Kolmogorov theory in regards to low
order statistical moments. In particular, it can be shown thatwhen
the energy spectrum follows a power law
 E(k) \propto k^{p},
with 1 p 3, the second order structure function has also a power
law, with the form
 \langle [\delta \mathbf{u}(r)]^2 \rangle \propto r^{p1} .
Since the experimental values obtained for the second order
structure function only deviate slightly from the 2/3 value
predicted by Kolmogorov theory, the value for
p is very
near to 5/3 (differences are about 2%). Thus the "Kolmogorov 5/3
spectrum" is generally observed in turbulence. However, for high
order structure functions the difference with the Kolmogorov
scaling is significant, and the breakdown of the statistical
selfsimilarity is clear. This behavior, and the lack of
universality of the C_n constants, are related with the phenomenon
of intermittency in turbulence. This is an important area of
research in this field, and a major goal of the modern theory of
turbulence is to understand what is really universal in the
inertial range.
See also
References and notes

http://www.weizmann.ac.il/home/fnfal/KRSPhysTodayApr2006.pdf
 http://www.eng.auburn.edu/users/thurobs/Turb.html
Turbulence
 http://www.fortunecity.com/emachines/e11/86/fluid.html
Turbulent Times for Fluids. It's important to notice that
turbulence is completely a different case from instability.
 U. Frisch. Turbulence: The Legacy of A. N. Kolmogorov.
Cambridge University Press, 1995.[1]
 J. Mathieu and J. Scott An Introduction to Turbulent
Flow. Cambridge University Press, 2000.
Further reading
General
 Falkovich, Gregory and Sreenivasan, Katepalli R. Lessons
from hydrodynamic turbulence, Physics Today, vol. 59, no. 4, pages
4349 (April 2006).[23683]
 U. Frisch. Turbulence: The Legacy of A. N.
Kolmogorov. Cambridge University Press, 1995.[23684]
 T. Bohr, M.H. Jensen, G. Paladin and A.Vulpiani. Dynamical
Systems Approach to Turbulence, Cambridge University Press,
1998.[23685]
 P.E. Dimotakis [23686] Highspeed digitalimage data
acquisition, processing, and Visualization system for turbulent
mixing and combustion 2007
Original scientific research papers
 , translated into English by
 , translated into English by
External links