Turbulence

# Turbulence

Overview
{{Other uses}} In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of flui
Discussion

Encyclopedia
{{Other uses}}
In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

property changes. This includes low momentum diffusion
Momentum diffusion
Momentum diffusion refers to the diffusion, or spread of momentum between particles of matter, usually in the liquid state....

, high momentum convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

, and rapid variation of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

and velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

in space and time. Nobel Laureate Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

described turbulence as "the most important unsolved problem of classical physics." Flow in which the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

dies out due to the action of fluid molecular viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

is called laminar flow
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...

. While there is no theorem relating Reynolds number (Re) to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In pipe flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 3000. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

due to boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition
Laminar-turbulent transition
The process of a laminar boundary layer becoming turbulent is known as boundary layer transition. This process is an extraordinarily complicated process which at present is not fully understood...

is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

of the fluid is decreased, or if the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the fluid is increased.

## Features

Turbulence is highly characterized by the following features: Irregularity: Turbulent flows are always highly irregular. This is why turbulence problems are always treated statistically rather than deterministically. Turbulent flow is always chaotic but not all chaotic flows are turbulent. Diffusivity
Diffusivity
Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum diffusivity...

: Turbulence is highly associated with rapid mixing. One of the useful effects of turbulence, it tends to accelerate the homogenization of any non-uniform fluid mixture. The process which brings any non-uniform state of a system into a uniform one is called mixing and when the system is in its uniform state, the system becomes a homogeneous system. A mixing process requires sufficient input of energy which is readily available in a turbulent flow. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is regarded as diffusivity. 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
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

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
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

's four-third power law and is governed by the random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula. A chaotic flow is never turbulent if it does not diffuse. For example, jet contrails are not turbulent as they don't diffuse even they are turbulent at the generation{{Citation needed|date=May 2011}}. Rotationality: Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching
Vortex stretching
In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum....

. 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 the structure function. 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 to the extent where their kinetic energy is overwhelmed by the fluid's molecular viscosity and dissipated into heat. This is why turbulence is always rotational and three dimensional. 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. Dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...

: To sustain turbulent flow, a constant source of energy supply is required. Otherwise, turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

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
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....

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
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

. Energy cascade: Turbulent flow can be realized as a superposition of a spectrum of velocity fluctuations and eddies on an over mean flow. The eddies are loosely defined as coherent patterns of 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 velocity fluctuations for each wave number. 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 length scales: Largest scales in the energy spectrum. These eddies obtain energy from the mean flow and also from each other. Thus these are the energy production eddies which contain the most of the energy. They have the large velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point 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. Kolmogorov length scales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

: 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 are in high frequency which is why turbulence is locally isotropic and homogeneous. Taylor microscale
Taylor microscale
The Taylor microscale is a length scale used to characterize a turbulent fluid flow. The Taylor microscale is the largest length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can...

s: The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor micro-scales are not dissipative scale but passes down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor micro-scales as a characteristic length scale and consider the energy cascade contains only the largest and smallest scales; while the later accommodate both the inertial sub-range and the viscous-sub layer. Nevertheless, Taylor micro-scales is often used in describing the term “turbulence” more conveniently as these Taylor micro-scales 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
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

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. Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n mathematician Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

) and the concept of self-similarity. As a result, the Kolmogorov microscales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified. Still, a complete description of turbulence remains one of the unsolved problems in physics
Unsolved problems in physics
This is a list of some of the major unsolved problems in physics. Some of these problems are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result...

. According to an apocryphal story, Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb
Horace Lamb
Sir Horace Lamb FRS was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics and Dynamical Theory of Sound...

(who had published a noted text book on Hydrodynamics)—his choice being quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

(instead of relativity) and turbulence. Lamb was quoted as saying in a speech to the British Association for the Advancement of Science
British Association for the Advancement of Science
frame|right|"The BA" logoThe British Association for the Advancement of Science or the British Science Association, formerly known as the BA, is a learned society with the object of promoting science, directing general attention to scientific matters, and facilitating interaction between...

, "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." A more detailed presentation of turbulence with emphasis on high-Reynolds number flow, intended for a general readership of physicists and applied mathematicians, is found in the Scholarpedia articles by R. Benzi and U. Frisch. and by G. Falkovich.

## Examples of turbulence

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• Smoke rising from a cigarette
Cigarette
A cigarette is a small roll of finely cut tobacco leaves wrapped in a cylinder of thin paper for smoking. The cigarette is ignited at one end and allowed to smoulder; its smoke is inhaled from the other end, which is held in or to the mouth and in some cases a cigarette holder may be used as well...

is turbulent flow
Flow
-Relating to the movement of material:* Fluid dynamics, or fluid flow, the motion of a gas or liquid* Environmental flow, the amount of water necessary in a watercourse to maintain a healthy ecosystem* Flow chemistry, a chemical reaction run in a continuous stream...

. For the first few centimeters, the flow is certainly laminar. Then smoke becomes turbulent as its Reynolds number increases, as its velocity and characteristic length are both increasing.
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• Flow over a golf ball
Golf ball
A golf ball is a ball designed to be used in the game of golf.Under the Rules of Golf, a golf ball weighs no more than 1.620 oz , has a diameter not less than 1.680 in , and performs within specified velocity, distance, and symmetry limits...

. (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.
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• The mixing of warm and cold air in the atmosphere by wind, which causes clear-air turbulence
Clear-Air Turbulence
Clear air turbulence is the turbulent movement of air masses in the absence of any visual cues such as clouds, and is caused when bodies of air moving at widely different speeds meet....

experienced during airplane flight, as well as poor astronomical seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

(the blurring of images seen through the atmosphere.)
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• Most of the terrestrial atmospheric circulation
Atmospheric circulation
Atmospheric circulation is the large-scale movement of air, and the means by which thermal energy is distributed on the surface of the Earth....

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• The oceanic and atmospheric mixed layer
Mixed layer
The oceanic or limnological mixed layer is a layer in which active turbulence has homogenized some range of depths. The surface mixed layer is a layer where this turbulence is generated by winds, cooling, or processes such as evaporation or sea ice formation which result in an increase in salinity...

s and intense oceanic currents.
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• The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubber
Scrubber
Scrubber systems are a diverse group of air pollution control devices that can be used to remove some particulates and/or gases from industrial exhaust streams. Traditionally, the term "scrubber" has referred to pollution control devices that use liquid to wash unwanted pollutants from a gas stream...

s, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engine
Internal combustion engine
The internal combustion engine is an engine in which the combustion of a fuel occurs with an oxidizer in a combustion chamber. In an internal combustion engine, the expansion of the high-temperature and high -pressure gases produced by combustion apply direct force to some component of the engine...

s and gas turbine
Gas turbine
A gas turbine, also called a combustion turbine, is a type of internal combustion engine. It has an upstream rotating compressor coupled to a downstream turbine, and a combustion chamber in-between....

s).
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• The external flow over all kind of vehicles such as cars, airplanes, ships and submarines.
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• The motions of matter in stellar atmospheres.
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• 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.
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• Race cars unable to follow each other through fast corners due to turbulence created by the leading car causing understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...

.
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• In windy conditions, trucks that are on the motorway gets buffeted by their wake.
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• Bridge supports (piers) in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds Number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.
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• In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structure activities and associated turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.
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• In the medical field of cardiology
Cardiology
Cardiology is a medical specialty dealing with disorders of the heart . The field includes diagnosis and treatment of congenital heart defects, coronary artery disease, heart failure, valvular heart disease and electrophysiology...

, a stethoscope is used to detect heart sounds
Heart sounds
Heart sounds, or heartbeats, are the noises generated by the beating heart and the resultant flow of blood through it...

and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis
Atherosclerosis
Atherosclerosis is a condition in which an artery wall thickens as a result of the accumulation of fatty materials such as cholesterol...

, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
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## Heat and momentum transfer

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

coefficient. Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual velocity $v=\left\left( \left\{\left\{v\right\}_\left\{x\right\}\right\},\left\{\left\{v\right\}_\left\{y\right\}\right\} \right\right)$ of every particle that passed through that point at any given time. Then one would find the actual velocity fluctuating about a mean value: $\left\{\left\{v\right\}_\left\{x\right\}\right\}=\underbrace\left\{\overline\right\}_\left\{\begin\left\{smallmatrix\right\} \text\left\{mean\right\} \\ \text\left\{value\right\} \end\left\{smallmatrix\right\}\right\}+\underbrace \left\{\left\{Other uses\right\}\right\}$
In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

property changes. This includes low momentum diffusion
Momentum diffusion
Momentum diffusion refers to the diffusion, or spread of momentum between particles of matter, usually in the liquid state....

, high momentum convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

, and rapid variation of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

and velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

in space and time. Nobel Laureate Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

described turbulence as "the most important unsolved problem of classical physics." Flow in which the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

dies out due to the action of fluid molecular viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

is called laminar flow
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...

. While there is no theorem relating Reynolds number (Re) to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In pipe flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 3000. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

due to boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition
Laminar-turbulent transition
The process of a laminar boundary layer becoming turbulent is known as boundary layer transition. This process is an extraordinarily complicated process which at present is not fully understood...

is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

of the fluid is decreased, or if the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the fluid is increased.

## Features

Turbulence is highly characterized by the following features: Irregularity: Turbulent flows are always highly irregular. This is why turbulence problems are always treated statistically rather than deterministically. Turbulent flow is always chaotic but not all chaotic flows are turbulent. Diffusivity
Diffusivity
Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum diffusivity...

: Turbulence is highly associated with rapid mixing. One of the useful effects of turbulence, it tends to accelerate the homogenization of any non-uniform fluid mixture. The process which brings any non-uniform state of a system into a uniform one is called mixing and when the system is in its uniform state, the system becomes a homogeneous system. A mixing process requires sufficient input of energy which is readily available in a turbulent flow. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is regarded as diffusivity. 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
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

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
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

's four-third power law and is governed by the random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula. A chaotic flow is never turbulent if it does not diffuse. For example, jet contrails are not turbulent as they don't diffuse even they are turbulent at the generation{{Citation needed|date=May 2011}}. Rotationality: Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching
Vortex stretching
In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum....

. 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 the structure function. 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 to the extent where their kinetic energy is overwhelmed by the fluid's molecular viscosity and dissipated into heat. This is why turbulence is always rotational and three dimensional. 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. Dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...

: To sustain turbulent flow, a constant source of energy supply is required. Otherwise, turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

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
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....

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
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

. Energy cascade: Turbulent flow can be realized as a superposition of a spectrum of velocity fluctuations and eddies on an over mean flow. The eddies are loosely defined as coherent patterns of 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 velocity fluctuations for each wave number. 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 length scales: Largest scales in the energy spectrum. These eddies obtain energy from the mean flow and also from each other. Thus these are the energy production eddies which contain the most of the energy. They have the large velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point 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. Kolmogorov length scales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

: 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 are in high frequency which is why turbulence is locally isotropic and homogeneous. Taylor microscale
Taylor microscale
The Taylor microscale is a length scale used to characterize a turbulent fluid flow. The Taylor microscale is the largest length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can...

s: The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor micro-scales are not dissipative scale but passes down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor micro-scales as a characteristic length scale and consider the energy cascade contains only the largest and smallest scales; while the later accommodate both the inertial sub-range and the viscous-sub layer. Nevertheless, Taylor micro-scales is often used in describing the term “turbulence” more conveniently as these Taylor micro-scales 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
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

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. Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n mathematician Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

) and the concept of self-similarity. As a result, the Kolmogorov microscales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified. Still, a complete description of turbulence remains one of the unsolved problems in physics
Unsolved problems in physics
This is a list of some of the major unsolved problems in physics. Some of these problems are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result...

. According to an apocryphal story, Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb
Horace Lamb
Sir Horace Lamb FRS was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics and Dynamical Theory of Sound...

(who had published a noted text book on Hydrodynamics)—his choice being quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

(instead of relativity) and turbulence. Lamb was quoted as saying in a speech to the British Association for the Advancement of Science
British Association for the Advancement of Science
frame|right|"The BA" logoThe British Association for the Advancement of Science or the British Science Association, formerly known as the BA, is a learned society with the object of promoting science, directing general attention to scientific matters, and facilitating interaction between...

, "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." A more detailed presentation of turbulence with emphasis on high-Reynolds number flow, intended for a general readership of physicists and applied mathematicians, is found in the Scholarpedia articles by R. Benzi and U. Frisch. and by G. Falkovich.

## Examples of turbulence

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• Smoke rising from a cigarette
Cigarette
A cigarette is a small roll of finely cut tobacco leaves wrapped in a cylinder of thin paper for smoking. The cigarette is ignited at one end and allowed to smoulder; its smoke is inhaled from the other end, which is held in or to the mouth and in some cases a cigarette holder may be used as well...

is turbulent flow
Flow
-Relating to the movement of material:* Fluid dynamics, or fluid flow, the motion of a gas or liquid* Environmental flow, the amount of water necessary in a watercourse to maintain a healthy ecosystem* Flow chemistry, a chemical reaction run in a continuous stream...

. For the first few centimeters, the flow is certainly laminar. Then smoke becomes turbulent as its Reynolds number increases, as its velocity and characteristic length are both increasing.
• NEWLINE
• Flow over a golf ball
Golf ball
A golf ball is a ball designed to be used in the game of golf.Under the Rules of Golf, a golf ball weighs no more than 1.620 oz , has a diameter not less than 1.680 in , and performs within specified velocity, distance, and symmetry limits...

. (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.
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• The mixing of warm and cold air in the atmosphere by wind, which causes clear-air turbulence
Clear-Air Turbulence
Clear air turbulence is the turbulent movement of air masses in the absence of any visual cues such as clouds, and is caused when bodies of air moving at widely different speeds meet....

experienced during airplane flight, as well as poor astronomical seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

(the blurring of images seen through the atmosphere.)
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• Most of the terrestrial atmospheric circulation
Atmospheric circulation
Atmospheric circulation is the large-scale movement of air, and the means by which thermal energy is distributed on the surface of the Earth....

• NEWLINE
• The oceanic and atmospheric mixed layer
Mixed layer
The oceanic or limnological mixed layer is a layer in which active turbulence has homogenized some range of depths. The surface mixed layer is a layer where this turbulence is generated by winds, cooling, or processes such as evaporation or sea ice formation which result in an increase in salinity...

s and intense oceanic currents.
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• The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubber
Scrubber
Scrubber systems are a diverse group of air pollution control devices that can be used to remove some particulates and/or gases from industrial exhaust streams. Traditionally, the term "scrubber" has referred to pollution control devices that use liquid to wash unwanted pollutants from a gas stream...

s, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engine
Internal combustion engine
The internal combustion engine is an engine in which the combustion of a fuel occurs with an oxidizer in a combustion chamber. In an internal combustion engine, the expansion of the high-temperature and high -pressure gases produced by combustion apply direct force to some component of the engine...

s and gas turbine
Gas turbine
A gas turbine, also called a combustion turbine, is a type of internal combustion engine. It has an upstream rotating compressor coupled to a downstream turbine, and a combustion chamber in-between....

s).
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• The external flow over all kind of vehicles such as cars, airplanes, ships and submarines.
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• The motions of matter in stellar atmospheres.
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• 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.
NEWLINE {{unsolved|physics|Is it possible to make a theoretical model to describe the behavior of a turbulent flow — in particular, its internal structures?}}NEWLINE
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• Race cars unable to follow each other through fast corners due to turbulence created by the leading car causing understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...

.
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• In windy conditions, trucks that are on the motorway gets buffeted by their wake.
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• Bridge supports (piers) in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds Number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.
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• In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structure activities and associated turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.
• NEWLINE
• In the medical field of cardiology
Cardiology
Cardiology is a medical specialty dealing with disorders of the heart . The field includes diagnosis and treatment of congenital heart defects, coronary artery disease, heart failure, valvular heart disease and electrophysiology...

, a stethoscope is used to detect heart sounds
Heart sounds
Heart sounds, or heartbeats, are the noises generated by the beating heart and the resultant flow of blood through it...

and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis
Atherosclerosis
Atherosclerosis is a condition in which an artery wall thickens as a result of the accumulation of fatty materials such as cholesterol...

, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
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## Heat and momentum transfer

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

coefficient. Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual velocity $v=\left\left( \left\{\left\{v\right\}_\left\{x\right\}\right\},\left\{\left\{v\right\}_\left\{y\right\}\right\} \right\right)$ of every particle that passed through that point at any given time. Then one would find the actual velocity fluctuating about a mean value: $\left\{\left\{v\right\}_\left\{x\right\}\right\}=\underbrace\left\{\overline\left\{\left\{\left\{v\right\}_\left\{x\right\}\right\}\right\}\right\}_\left\{\begin\left\{smallmatrix\right\} \text\left\{mean\right\} \\ \text\left\{value\right\} \end\left\{smallmatrix\right\}\right\}+\underbrace \left\{\left\{Other uses\right\}\right\}$
In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

property changes. This includes low momentum diffusion
Momentum diffusion
Momentum diffusion refers to the diffusion, or spread of momentum between particles of matter, usually in the liquid state....

, high momentum convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

, and rapid variation of pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

and velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

in space and time. Nobel Laureate Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

described turbulence as "the most important unsolved problem of classical physics." Flow in which the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

dies out due to the action of fluid molecular viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

is called laminar flow
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...

. While there is no theorem relating Reynolds number (Re) to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In pipe flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 3000. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

due to boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition
Laminar-turbulent transition
The process of a laminar boundary layer becoming turbulent is known as boundary layer transition. This process is an extraordinarily complicated process which at present is not fully understood...

is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

of the fluid is decreased, or if the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the fluid is increased.

## Features

Turbulence is highly characterized by the following features: Irregularity: Turbulent flows are always highly irregular. This is why turbulence problems are always treated statistically rather than deterministically. Turbulent flow is always chaotic but not all chaotic flows are turbulent. Diffusivity
Diffusivity
Diffusivity can refer to:*Diffusivity of heat*Diffusivity of mass:** Molecular diffusivity ** Eddy diffusivity*Momentum diffusivity...

: Turbulence is highly associated with rapid mixing. One of the useful effects of turbulence, it tends to accelerate the homogenization of any non-uniform fluid mixture. The process which brings any non-uniform state of a system into a uniform one is called mixing and when the system is in its uniform state, the system becomes a homogeneous system. A mixing process requires sufficient input of energy which is readily available in a turbulent flow. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is regarded as diffusivity. 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
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

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
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

's four-third power law and is governed by the random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula. A chaotic flow is never turbulent if it does not diffuse. For example, jet contrails are not turbulent as they don't diffuse even they are turbulent at the generation{{Citation needed|date=May 2011}}. Rotationality: Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching
Vortex stretching
In fluid dynamics, vortex stretching is the lengthening of vortices in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum....

. 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 the structure function. 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 to the extent where their kinetic energy is overwhelmed by the fluid's molecular viscosity and dissipated into heat. This is why turbulence is always rotational and three dimensional. 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. Dissipation
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...

: To sustain turbulent flow, a constant source of energy supply is required. Otherwise, turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

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
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....

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
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

. Energy cascade: Turbulent flow can be realized as a superposition of a spectrum of velocity fluctuations and eddies on an over mean flow. The eddies are loosely defined as coherent patterns of 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 velocity fluctuations for each wave number. 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 length scales: Largest scales in the energy spectrum. These eddies obtain energy from the mean flow and also from each other. Thus these are the energy production eddies which contain the most of the energy. They have the large velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point 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. Kolmogorov length scales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

: 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 are in high frequency which is why turbulence is locally isotropic and homogeneous. Taylor microscale
Taylor microscale
The Taylor microscale is a length scale used to characterize a turbulent fluid flow. The Taylor microscale is the largest length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can...

s: The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor micro-scales are not dissipative scale but passes down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor micro-scales as a characteristic length scale and consider the energy cascade contains only the largest and smallest scales; while the later accommodate both the inertial sub-range and the viscous-sub layer. Nevertheless, Taylor micro-scales is often used in describing the term “turbulence” more conveniently as these Taylor micro-scales 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
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

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. Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n mathematician Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

) and the concept of self-similarity. As a result, the Kolmogorov microscales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified. Still, a complete description of turbulence remains one of the unsolved problems in physics
Unsolved problems in physics
This is a list of some of the major unsolved problems in physics. Some of these problems are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result...

. According to an apocryphal story, Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

God
God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb
Horace Lamb
Sir Horace Lamb FRS was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics and Dynamical Theory of Sound...

(who had published a noted text book on Hydrodynamics)—his choice being quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

(instead of relativity) and turbulence. Lamb was quoted as saying in a speech to the British Association for the Advancement of Science
British Association for the Advancement of Science
frame|right|"The BA" logoThe British Association for the Advancement of Science or the British Science Association, formerly known as the BA, is a learned society with the object of promoting science, directing general attention to scientific matters, and facilitating interaction between...

, "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." A more detailed presentation of turbulence with emphasis on high-Reynolds number flow, intended for a general readership of physicists and applied mathematicians, is found in the Scholarpedia articles by R. Benzi and U. Frisch. and by G. Falkovich.

## Examples of turbulence

NEWLINE
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• Smoke rising from a cigarette
Cigarette
A cigarette is a small roll of finely cut tobacco leaves wrapped in a cylinder of thin paper for smoking. The cigarette is ignited at one end and allowed to smoulder; its smoke is inhaled from the other end, which is held in or to the mouth and in some cases a cigarette holder may be used as well...

is turbulent flow
Flow
-Relating to the movement of material:* Fluid dynamics, or fluid flow, the motion of a gas or liquid* Environmental flow, the amount of water necessary in a watercourse to maintain a healthy ecosystem* Flow chemistry, a chemical reaction run in a continuous stream...

. For the first few centimeters, the flow is certainly laminar. Then smoke becomes turbulent as its Reynolds number increases, as its velocity and characteristic length are both increasing.
• NEWLINE
• Flow over a golf ball
Golf ball
A golf ball is a ball designed to be used in the game of golf.Under the Rules of Golf, a golf ball weighs no more than 1.620 oz , has a diameter not less than 1.680 in , and performs within specified velocity, distance, and symmetry limits...

. (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.
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• The mixing of warm and cold air in the atmosphere by wind, which causes clear-air turbulence
Clear-Air Turbulence
Clear air turbulence is the turbulent movement of air masses in the absence of any visual cues such as clouds, and is caused when bodies of air moving at widely different speeds meet....

experienced during airplane flight, as well as poor astronomical seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

(the blurring of images seen through the atmosphere.)
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• Most of the terrestrial atmospheric circulation
Atmospheric circulation
Atmospheric circulation is the large-scale movement of air, and the means by which thermal energy is distributed on the surface of the Earth....

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• The oceanic and atmospheric mixed layer
Mixed layer
The oceanic or limnological mixed layer is a layer in which active turbulence has homogenized some range of depths. The surface mixed layer is a layer where this turbulence is generated by winds, cooling, or processes such as evaporation or sea ice formation which result in an increase in salinity...

s and intense oceanic currents.
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• The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubber
Scrubber
Scrubber systems are a diverse group of air pollution control devices that can be used to remove some particulates and/or gases from industrial exhaust streams. Traditionally, the term "scrubber" has referred to pollution control devices that use liquid to wash unwanted pollutants from a gas stream...

s, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engine
Internal combustion engine
The internal combustion engine is an engine in which the combustion of a fuel occurs with an oxidizer in a combustion chamber. In an internal combustion engine, the expansion of the high-temperature and high -pressure gases produced by combustion apply direct force to some component of the engine...

s and gas turbine
Gas turbine
A gas turbine, also called a combustion turbine, is a type of internal combustion engine. It has an upstream rotating compressor coupled to a downstream turbine, and a combustion chamber in-between....

s).
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• The external flow over all kind of vehicles such as cars, airplanes, ships and submarines.
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• The motions of matter in stellar atmospheres.
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• 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.
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• Race cars unable to follow each other through fast corners due to turbulence created by the leading car causing understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...

.
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• In windy conditions, trucks that are on the motorway gets buffeted by their wake.
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• Bridge supports (piers) in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds Number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.
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• In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structure activities and associated turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.
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• In the medical field of cardiology
Cardiology
Cardiology is a medical specialty dealing with disorders of the heart . The field includes diagnosis and treatment of congenital heart defects, coronary artery disease, heart failure, valvular heart disease and electrophysiology...

, a stethoscope is used to detect heart sounds
Heart sounds
Heart sounds, or heartbeats, are the noises generated by the beating heart and the resultant flow of blood through it...

and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis
Atherosclerosis
Atherosclerosis is a condition in which an artery wall thickens as a result of the accumulation of fatty materials such as cholesterol...

, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
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## Heat and momentum transfer

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

coefficient. Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual velocity $v=\left\left( \left\{\left\{v\right\}_\left\{x\right\}\right\},\left\{\left\{v\right\}_\left\{y\right\}\right\} \right\right)$ of every particle that passed through that point at any given time. Then one would find the actual velocity fluctuating about a mean value: $\left\{\left\{v\right\}_\left\{x\right\}\right\}=\underbrace\left\{\overline\left\{\left\{\left\{v\right\}_\left\{x\right\}\right\}\right\}\right\}_\left\{\begin\left\{smallmatrix\right\} \text\left\{mean\right\} \\ \text\left\{value\right\} \end\left\{smallmatrix\right\}\right\}+\underbrace\left\{\left\{\left\{\left\{\left\{v\right\}\text{'}\right\}\right\}_\left\{x\right\}\right\}\right\}_\left\{\text\left\{fluctuation\right\}\right\}\text\left\{ \right\}\text\left\{, \right\}\left\{\left\{v\right\}_\left\{y\right\}\right\}=\overline\left\{\left\{\left\{v\right\}_\left\{y\right\}\right\}\right\}+\left\{\left\{\left\{v\right\}\text{'}\right\}_\left\{y\right\}\right\}$ and similarly for temperature $\left\left( T=\overline\left\{T\right\}+\left\{T\right\}\text{'} \right\right)$ and pressure $\left\left( P=\overline\left\{P\right\}+\left\{P\right\}\text{'} \right\right)$, where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables. The heat flux and momentum transfer (represented by the shear stress $\tau$) in the direction normal to the flow for a given time are \begin\left\{align\right\} & q=\underbrace\left\{\left\{\left\{\left\{\left\{v\right\}\text{'}\right\}\right\}_\left\{y\right\}\right\}\rho \left\{\left\{c\right\}_\left\{P\right\}\right\}\left\{T\right\}\text{'}\right\}_\left\{\text\left\{experimental value\right\}\right\}=-\left\{\left\{k\right\}_\left\{\text\left\{turb\right\}\right\}\right\}\frac\left\{\partial \overline\left\{T\right\}\right\}\left\{\partial y\right\} \\ & \tau =\underbrace\left\{-\rho \overline\left\{\left\{\left\{\left\{\left\{v\right\}\text{'}\right\}\right\}_\left\{y\right\}\right\}\left\{\left\{\left\{\left\{v\right\}\text{'}\right\}\right\}_\left\{x\right\}\right\}\right\}\right\}_\left\{\text\left\{experimental value\right\}\right\}=\left\{\left\{\mu \right\}_\left\{\text\left\{turb\right\}\right\}\right\}\frac\left\{\partial \overline\left\{\left\{\left\{v\right\}_\left\{x\right\}\right\}\right\}\right\}\left\{\partial y\right\} \\ \end\left\{align\right\} where $\left\{\left\{c\right\}_\left\{P\right\}\right\}$ is the heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

at constant pressure, $\rho$ is the density of the fluid, $\left\{\left\{\mu \right\}_\left\{\text\left\{turb\right\}\right\}\right\}$ is the coefficient of turbulent viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

and $\left\{\left\{k\right\}_\left\{\text\left\{turb\right\}\right\}\right\}$ is the turbulent thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

.

## 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 $\eta = \left\left(\frac\left\{\nu^3\right\}\left\{\varepsilon\right\}\right\right)^\left\{1/4\right\}$. This is today known as the Kolmogorov length scale (see Kolmogorov microscales
Kolmogorov microscales
Kolmogorov microscales are the smallest scales in turbulent flow. They are defined bywhere \epsilon is the average rate of energy dissipation per unit mass, and \nu is the kinematic viscosity of the fluid....

). 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 $E\left(k\right)$, where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x): $\mathbf\left\{u\right\}\left(\mathbf\left\{x\right\}\right) = \iiint_\left\{\mathbb\left\{R\right\}^3\right\} \widehat\left\{\mathbf\left\{u\right\}\right\}\left(\mathbf\left\{k\right\}\right)e^\left\{i \mathbf\left\{k \cdot x\right\}\right\} \mathrm\left\{d\right\}^3\mathbf\left\{k\right\}$, where û(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, $\frac\left\{1\right\}\left\{2\right\}\langle u_i u_i \rangle = \int_\left\{0\right\}^\left\{\infty\right\}E\left(k\right)\mathrm\left\{d\right\}k$, where $1/2\langle u_i u_i \rangle$ is the mean turbulent kinetic energy of the flow. 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 $E\left(k\right) = C \varepsilon^\left\{2/3\right\} k^\left\{-5/3\right\}$, 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: $\delta \mathbf\left\{u\right\}\left(r\right) = \mathbf\left\{u\right\}\left(\mathbf\left\{x\right\} + \mathbf\left\{r\right\}\right) - \mathbf\left\{u\right\}\left(\mathbf\left\{x\right\}\right)$; 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 $\lambda$, $\delta \mathbf\left\{u\right\}\left(\lambda r\right)$ should have the same statistical distribution as $\lambda^\left\{\beta\right\}\delta \mathbf\left\{u\right\}\left(r\right)$, 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 $\langle \left[\delta \mathbf\left\{u\right\}\left(r\right)\right]^n \rangle = C_n \varepsilon^\left\{n/3\right\} r^\left\{n/3\right\}$, 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 $E\left(k\right) \propto k^\left\{-p\right\}$, with $1 < p < 3$, the second order structure function has also a power law, with the form $\langle \left[\delta \mathbf\left\{u\right\}\left(r\right)\right]^2 \rangle \propto r^\left\{p-1\right\}$. 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.

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• Astronomical seeing
Astronomical seeing
Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

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• Atmospheric dispersion modeling
Atmospheric dispersion modeling
Atmospheric dispersion modeling is the mathematical simulation of how air pollutants disperse in the ambient atmosphere. It is performed with computer programs that solve the mathematical equations and algorithms which simulate the pollutant dispersion...

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• Chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

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• Clear-air turbulence
Clear-Air Turbulence
Clear air turbulence is the turbulent movement of air masses in the absence of any visual cues such as clouds, and is caused when bodies of air moving at widely different speeds meet....

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• Constructal theory
Constructal theory
The constructal law puts forth the idea that the generation of design in nature is a physics phenomenon that unites all animate and inanimate systems, and that this phenomenon is covered by the Constructal Law...

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• Downdrafts
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• Eddy covariance
Eddy covariance
The eddy covariance technique is a key atmospheric flux measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers...

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• Fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

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• Darcy–Weisbach equation
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• Eddy
Eddy (fluid dynamics)
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid flows past an obstacle. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object...

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• Navier-Stokes equations
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

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• Large eddy simulation
Large eddy simulation
Large eddy simulation is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and many of the issues unique to LES were first explored by Deardorff...

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• Poiseuille's law
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• Lagrangian coherent structure
Lagrangian coherent structure
Lagrangian coherent structures are structures which separate dynamically distinct regions in time-varying systems such as turbulent flows in fluid mechanics...

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• Turbulence kinetic energy
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• Mesocyclone
Mesocyclone
A mesocyclone is a vortex of air, approximately 2 to 10 miles in diameter , within a convective storm....

s
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• Reynolds Number
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• Swing bowling
Swing bowling
Swing bowling is a technique used for bowling in the sport of cricket. Practitioners are known as swing bowlers. Swing bowling is generally classed as a subtype of fast bowling.-Physics of swing bowling:...

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• Taylor microscale
Taylor microscale
The Taylor microscale is a length scale used to characterize a turbulent fluid flow. The Taylor microscale is the largest length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can...

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• Turbulence modeling
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• Velocimetry
Velocimetry
Velocimetry is the measurement of the velocity of fluids, as often used to solve fluid dynamics problems, or to study fluid networks, as well as in industrial and process control applications, or in the creation of new kinds of fluid flow sensors...

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• Vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

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• Vortex generator
Vortex generator
A vortex generator is an aerodynamic surface, consisting of a small vane or bump that creates a vortex. Vortex generators can be found on many devices, but the term is most often used in aircraft design....

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• Wake turbulence
Wake turbulence
Wake turbulence is turbulence that forms behind an aircraft as it passes through the air. This turbulence includes various components, the most important of which are wing vorticies and jetwash. Jetwash refers simply to the rapidly moving gases expelled from a jet engine; it is extremely turbulent,...

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• Wave turbulence
Wave turbulence
Wave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...

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• Wingtip vortices
Wingtip vortices
Wingtip vortices are tubes of circulating air that are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure...

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• Wind tunnel
Wind tunnel
A wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.-Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...

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### Original scientific research papers and classical monographs

{{ru icon}}, translated into English by {{cite journal|last=Kolmogorov|first=Andrey Nikolaevich|date=July 8, 1991|title=The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers|journal=Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences|volume=434|issue=1991|pages=9–13|doi=10.1098/rspa.1991.0075|bibcode = 1991RSPSA.434....9K }} {{ru icon}}, translated into English by {{cite journal|last=Kolmogorov|first=Andrey Nikolaevich|date=July 8, 1991|title=The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers|journal=Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences|volume=434|issue=1980|pages=15–17|doi=10.1098/rspa.1991.0076|bibcode = 1991RSPSA.434...15K }}NEWLINE
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• G. K. Batchelor, The theory of homogeneous turbulence. Cambridge University Press, 1953.
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