Turbulence

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.

[5] The turbulence intensity affects many fields, for examples fish ecology,[6] air pollution,[7] precipitation,[8] and climate change.

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.

Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that a statistical description is needed.

The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity.

"[20][a] 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.

"[21][22] 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.

[23] 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.

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.

The heat flux and momentum transfer (represented by the shear stress τ) in the direction normal to the flow for a given time are where cP is the heat capacity at constant pressure, ρ is the density of the fluid, μturb is the coefficient of turbulent viscosity and kturb is the turbulent thermal conductivity.

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers, the small-scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned).

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity ν and the rate of energy dissipation ε.

For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function E(k), where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x): where û(k) is the Fourier transform of the flow velocity field.

This is one of the most famous results of Kolmogorov 1941 theory,[26] describing transport of energy through scale space without any loss or gain.

The Kolmogorov five-thirds law was first observed in a tidal channel,[27] and considerable experimental evidence has since accumulated that supports it.

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 Cn 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 r.[31] 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.