Editing Turbulence (section)

==Features==
[[File:False color image of the far field of a submerged turbulent jet.jpg|thumb|right|Flow visualization of a turbulent jet, made by [[Planar laser-induced fluorescence|laser-induced fluorescence]]. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.]]
Turbulence is characterized by the following features:

; Irregularity : Turbulent flows are always highly irregular. For this reason, turbulence problems are normally treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent.

; {{anchor|Diffusivity}}Diffusivity :The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures.  The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".<ref name="Ferziger Peric 2012 p.265-307">{{cite book | last1=Ferziger | first1=Joel H. | last2=Peric | first2=Milovan | title=Computational Methods for Fluid Dynamics | publisher=Springer Science & Business Media | date=6 December 2012 | isbn=978-3-642-56026-2 | ol=27025861M | oclc=725390736 | pages=265–307}}</ref>

''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 [[Lewis Fry Richardson|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.

;[[Rotationality]] :Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as [[vortex stretching]]. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish and maintain identifiable structure function.<ref name="Kundu, Pijush K. 2012 pp. 537">Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (2012). ''Fluid Mechanics''. Netherlands: Elsevier Inc. pp. 537–601. {{ISBN|978-0-12-382100-3}}.</ref> In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow is always rotational and three dimensional.<ref name="Kundu, Pijush K. 2012 pp. 537"/> For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent. On the other hand, oceanic flows are dispersive but essentially non rotational and therefore are not turbulent.<ref name="Kundu, Pijush K. 2012 pp. 537"/>

;[[Dissipation]] : To sustain turbulent flow, a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of [[Eddy (fluid dynamics)|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 flow|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 microscales|Kolmogorov length scale]].

Via this [[energy cascade]], turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a [[mean flow]]. The eddies are loosely defined as coherent patterns of flow velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in flow velocity fluctuations for each length scale ([[wavenumber]]). The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.
; Integral time scale
The integral time scale for a Lagrangian flow can be defined as:

: <math>T = \left ( \frac{1}{\langle u'u'\rangle} \right )\int_0^\infty \langle u'u'(\tau)\rangle \, d\tau</math>

where ''u''&prime; is the velocity fluctuation, and <math>\tau</math> is the time lag between measurements.<ref name="Tennekes 1972">{{cite book | last1=Tennekes | first1=Hendrik | last2=Lumley | first2=John L.| title=A First Course in Turbulence | publisher=MIT Press | publication-place=Cambridge, Mass. | date=1972 | isbn=978-0-262-20019-6}}</ref>
; Integral length scales 
: Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly [[anisotropic]] and are defined in terms of the normalized two-point flow velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as
:: <math>L = \left ( \frac{1}{\langle u'u'\rangle} \right ) \int_0^\infty \langle u'u'(r)\rangle \, dr</math>
: where ''r'' is the distance between two measurement locations, and ''u''&prime; is the velocity fluctuation in that same direction.<ref name="Tennekes 1972"/>
; [[Kolmogorov microscales|Kolmogorov length scales]] : Smallest scales in the spectrum that form the viscous sub-layer range. In this range, the energy input from nonlinear interactions and the energy drain from viscous dissipation are in exact balance. The small scales have high frequency, causing turbulence to be locally [[isotropic]] and homogeneous.
; [[Taylor microscale]]s : The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scales, but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.

Although it is possible to find some particular solutions of the [[Navier–Stokes equations]] governing fluid motion, all such solutions are unstable to finite perturbations 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. The [[Russia]]n mathematician [[Andrey Kolmogorov]] proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by [[Lewis Fry Richardson|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.<ref>{{cite journal|first1=Gregory|last1=Falkovich|first2=K. R.|last2=Sreenivasan|title=Lessons from hydrodynamic turbulence|journal=[[Physics Today]]|volume=59|issue=4|pages=43–49|date=April 2006| doi=10.1063/1.2207037|bibcode=2006PhT....59d..43F |url=http://www.weizmann.ac.il/home/fnfal/papers/PhysToday.pdf |via=weizmann.ac.il }}</ref>

A complete description of turbulence is 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 [[Theory of relativity|relativity]]? And why turbulence? I really believe he will have an answer for the first."<ref>{{cite book|last=Marshak|first=Alex|title=3D radiative transfer in cloudy atmospheres|page=76|url=https://books.google.com/books?id=wzg6wnpHyCUC|year=2005|publisher=[[Springer Science+Business Media|Springer]]|isbn=978-3-540-23958-1}}</ref>{{refn |group=lower-alpha |The story has also been attributed to [[John von Neumann]], [[Arnold Sommerfeld]], [[Theodore von Kármán]], and [[Albert Einstein]].}} A similar witticism has been attributed to [[Horace Lamb]] 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 more optimistic."<ref>{{cite journal|last=Mullin|first=Tom|date=11 November 1989|title=Turbulent times for fluids|journal=[[New Scientist]]}}</ref><ref>{{cite book|last=Davidson|first=P. A.|title=Turbulence: An Introduction for Scientists and Engineers|url=https://books.google.com/books?id=rkOmKzujZB4C&q=%22when+I+die+and+go+to+Heaven+there+are+two+matters+on+which+I+hope+for+enlightenment%22&pg=PA24|year=2004|publisher=[[Oxford University Press]]|isbn=978-0-19-852949-1}}</ref>