Turbulence

In fluid dynamics, turbulence or turbulent flow is fluid motion exhibiting chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between those layers.

Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases.

The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Physicist Richard Feynman described turbulence as the most important unsolved problem in classical physics.

The turbulence intensity affects many fields, for examples fish ecology, air pollution, precipitation, and climate change.

Examples of turbulence

thumb|right|[[Laminar flow|Laminar and turbulent water flow over the hull of a submarine. As the relative velocity of the water increases, turbulence occurs.]]

thumb|right|Turbulence in the [[Wingtip vortices|tip vortex from an airplane wing passing through coloured smoke ]]

Smoke rising from a cigarette. For the first few centimeters, the smoke is laminar. The smoke plume becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic length scale.
Flow over a golf ball. (This can be best understood by considering the golf ball to be stationary, with air flowing over it.) If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable (pressure decreasing in the flow direction) to unfavorable (pressure increasing in the flow direction), creating a large region of low pressure behind the ball that creates high form drag. To prevent this, the surface is dimpled to perturb the boundary layer and promote turbulence. This results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.
Clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing (the blurring of images seen through the atmosphere).
Most of the terrestrial atmospheric circulation.
The oceanic and atmospheric mixed layers and intense oceanic currents.
The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubbers, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engines and gas turbines).
The external flow over all kinds of vehicles such as cars, airplanes, ships, and submarines.
The motions of matter in stellar atmospheres.
A jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence.
Biologically generated turbulence resulting from swimming animals affects ocean mixing.
Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.
Bridge supports (piers) in water. When river flow is slow, water flows smoothly around the support legs. 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.
In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structures and 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.

In the medical field of cardiology, a stethoscope is used to detect heart sounds 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, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
Recently, turbulence in porous media became a highly debated subject.
Strategies used by animals for olfactory navigation, and their success, are heavily influenced by turbulence affecting the odor plume.
Pyroclastic flow, one of the deadliest and most destructive volcanic eruption hazards consisting of hot, fast-moving gas and volcanic matter, experiences turbulence. Made up of chemically diverse components, pyroclastic flows have heterogeneous distributions in physical features such as temperature, density, and flow velocity. These differences promote turbulent mixing within the flows themselves as well as with the external environment as they move down volcanic slopes. Turbulent mixing eventually allows for greater homogeneity within pyroclastic flows.

Features

thumb|right|Flow visualization of a turbulent jet, made by [[Planar laser-induced fluorescence|laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.]]

Turbulence is characterized by the following features:

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

; Diffusivity :The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".

::Turbulent diffusion is usually described by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson's four-third power law and is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.

;Rotationality :Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish and maintain identifiable structure function. In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow is always rotational and three dimensional.

; Integral length scales

: Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point flow velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as <math display=block>L = \frac{1}{\langle u'u'\rangle} \int_0^\infty \langle u'u'(r)\rangle \, dr</math> where r is the distance between two measurement locations, and u′ is the velocity fluctuation in that same direction.

A complete description of turbulence is one of the unsolved problems in physics. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb in a speech to the British Association for the Advancement of Science: "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather more optimistic."

Onset of turbulence

thumb|right|The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place

The onset of turbulence can be, to some extent, predicted by the Reynolds number, which is the ratio of inertial forces to viscous forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.

This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of Reynolds numbers to both situations allows scaling factors to be developed.

A flow situation in which the kinetic energy is significantly absorbed due to the action of fluid molecular viscosity gives rise to a laminar flow regime. For this the dimensionless quantity the Reynolds number () is used as a guide.

With respect to laminar and turbulent flow regimes:

laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The Reynolds number is defined as

:<math> \mathrm{Re} = \frac{\rho v L}{\mu} \,,</math>

where:

is the density of the fluid (SI units: kg/m<sup>3</sup>)
is a characteristic velocity of the fluid with respect to the object (m/s)
is a characteristic linear dimension (m)
is the dynamic viscosity of the fluid (Pa·s or N·s/m<sup>2</sup> or kg/(m·s)).

While there is no theorem directly relating the non-dimensional Reynolds number 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 Poiseuille 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 4000.

The transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.

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 coefficient.

Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:

:<math>v_x = \underbrace{\overline{v}_x}_\text{mean value} + \underbrace{v'_x}_\text{fluctuation} \quad \text{and} \quad v_y=\overline{v}_y + v'_y \,;</math>

and similarly for temperature () and pressure (), 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 ) in the direction normal to the flow for a given time are

:<math>\begin{align}

q&=\underbrace{v'_y \rho c_P T'}_\text{experimental value} = -k_\text{turb}\frac{\partial \overline{T{\partial y} \,; \\

\tau &=\underbrace{-\rho \overline{v'_y v'_x_\text{experimental value} = \mu_\text{turb}\frac{\partial \overline{v}_x}{\partial y} \,;

\end{align}</math>

where is the heat capacity at constant pressure, is the density of the fluid, is the coefficient of turbulent viscosity and is the turbulent thermal conductivity. describing transport of energy through scale space without any loss or gain. The Kolmogorov five-thirds law was first observed in a tidal channel, and considerable experimental evidence has since accumulated that supports it.

Outside of the inertial area, one can find the formula below :

:<math>E(k) = K_0 \varepsilon^\frac23 k^{-\frac53} \exp \left[ - \frac{3 K_0}{2} \left( \frac{\nu^3 k^4}{\varepsilon} \right)^{\frac13} \right] \,,</math>

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 and non-intermittent in the inertial range. A usual way of studying turbulent flow velocity fields is by means of flow velocity increments:

:<math>\delta \mathbf{u}(r) = \mathbf{u}(\mathbf{x} + \mathbf{r}) - \mathbf{u}(\mathbf{x}) \,;</math>

that is, the difference in flow velocity between points separated by a vector (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of ). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation when statistics are computed. The statistical scale-invariance without intermittency implies that the scaling of flow velocity increments should occur with a unique scaling exponent , so that when is scaled by a factor ,

:<math>\delta \mathbf{u}(\lambda r)</math>

should have the same statistical distribution as

:<math>\lambda^\beta \delta \mathbf{u}(r)\,,</math>

with independent of the scale . From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as structure functions in turbulence) should scale as

:<math>\Big\langle \big ( \delta \mathbf{u}(r)\big )^n \Big\rangle = C_n \langle (\varepsilon r )^\frac{n}{3} \rangle \,,</math>

where the brackets denote the statistical average, and the would be universal constants.

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the value predicted by the theory, becoming a non-linear function of the order of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov 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

:<math>E(k) \propto k^{-p} \,,</math>

with , the second order structure function has also a power law, with the form

:<math>\Big\langle \big (\delta \mathbf{u}(r)\big )^2 \Big\rangle \propto r^{p-1} \,,</math>

Since the experimental values obtained for the second order structure function only deviate slightly from the value predicted by Kolmogorov theory, the value for is very near to (differences are about 2%). Thus the "Kolmogorov − 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 constants, are related with the phenomenon of intermittency in turbulence and can be related to the non-trivial scaling behavior of the dissipation rate averaged over scale . 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 universal in the inertial range, and how to deduce intermittency properties from the Navier-Stokes equations, i.e. from first principles.

Notes

References

External links

Center for Turbulence Research, Scientific papers and books on turbulence
Center for Turbulence Research, Stanford University
Scientific American article
Air Turbulence Forecast
international CFD database iCFDdatabase
Fluid Mechanics website with movies, Q&A, etc
Johns Hopkins public database with direct numerical simulation data
TurBase public database with experimental data from European High Performance Infrastructures in Turbulence (EuHIT)