Turbulence
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. The (dimensionless) Reynolds number characterizes whether flow conditions lead to laminar or turbulent flow; e.g. for pipe flow, a Reynolds number above about 4000 (A Reynolds number between 2100 and 4000 is known as transitional flow) will be turbulent. At very low speeds the flow is laminar, i.e., the flow is smooth (though it may involve vortices on a large scale). As the speed increases, at some point the transition is made to turbulent flow. 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. Because laminar-turbulent transition is 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.
In two dimensional turbulence (as can be approximated in the atmosphere or ocean), energy actually flows to larger scales.  This is referred to as the inverse energy cascade and is characterized by a k^{-(5/3)} in the power spectrum.  This is the main reason why large scale weather features such as hurricanes occur.
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 four-third 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 Navier-Stokes 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 self-similarity.  As a result,  the Kolmogorov microscales were named after him. It is now known that the self-similarity 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 mechanics (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
- 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 clear-air 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, 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 1941 Theory
The 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 = (\frac{\nu^3}{\varepsilon})^{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 wavenumber vector 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 u(k) is the Fourier transform of the velocity field. Thus, E(k)dk represents the contribution to the kinetic energy from all the Fourier modes with k < |k| < k + dk, 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 self-similar at different scales. This essentially means that the statistics are scale-invariant 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 scale-invariance 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 than
\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 ^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 non-linear 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 that 
when the energy spectrum follows a power law
E(k) \propto k^{-p},
with 1 , the second order structure function has also a power law, with the form
\langle ^2  \rangle \propto r^{p-1} .
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 self-similarity 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.