**Turbulence**, a scientific term to describe certain complex and unpredictable motions of a fluid, is part of our daily experience and has been for a long time. No telescope or microscope is needed to contemplate the volutes of smoke from a cigarette, the elegant arabesques of cream poured into coffee and the vigorous eddies of a mountain stream. In an airplane we sometimes experience bursts of “clear air turbulence”. Ultrasonography can reveal turbulent blood flow in our arteries; satellite pictures may show turbulent meteorological perturbations; computer simulations reveal turbulent fluctuations of mass in the Universe on scales of tens of megaparsecs. Without turbulence, urban pollution would linger around for centuries, the heat produced by nuclear reactions in the interior of stars would not be able to escape on an acceptable time scale and meteorological phenomena would be predictable almost for ever.

Actually the word “turbulence” (Latin: turbulentia) originally refers to the disorderly motion of a crowd (turba). In the Middle Ages it was frequently used to mean just “trouble”, a word which derives from it. Even today “turbulent” may refer to social or personal behaviour. Its scientific usage refers to irregular and seemingly random motion of a fluid. This definition, which is far from exhaustive, tries to express in a synthetic way one of the most complex and fascinating phenomenon of natural science, from Antiquity to present days.

The subject has indeed a very long history. Lucretius described eddy motion in his De rerum natura. Subsequently, Leonardo was probably the first to use the word turbulence (in Italian turbolenza) with its modern meaning and to observe the slow decay of eddies formed behind the pillars of a bridge. Next, Euler wrote the equations of incompressible ideal or inviscid (zero-viscosity) flow in both two and three dimensions and realized the importance of vorticity. Years later Navier generalized these equation to include viscosity. Because of further work by Stokes, the equations are known as the Navier–Stokes equation.

In fluid dynamics, **turbulence** or **turbulent flow** is a fluid flow regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. A flow that is not turbulent is called laminar flow. The (dimensionless) Reynolds number characterizes whether flow conditions lead to laminar or turbulent flow.

The flow of water over a simple smooth object, such as a sphere, illustrates it. At very low speeds the flow is laminar; i.e., the flow is smooth (though it may involve vortices on a large scale). As the speed increases, at some point, the transition is made to turbulent (“chaotic”) flow. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases.

The structure and location of boundary layer separation often change, sometimes resulting in a reduction of overall drag. Because the laminar-turbulent transition is governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.

## Examples of turbulent motions

- Smoke produced by a cigarette, which changes from laminar to turbulent as the velocity and characteristic length scale of the flow increase.
- Turbulence in the upper atmosphere, which can cause the phenomenon of astronomical seeing.
- Much of the Earth’s atmospheric circulation.
- The mixed layer in oceans and atmosphere, as well as intense ocean currents.
- Flows inside many industrial machines and equipment (pipelines, internal combustion engines, gas turbines, etc.).
- The flows outside of all types of moving vehicles (cars, planes, trains, ships, etc.).
- Motions within stellar atmospheres.
- Swimming animals can generate turbulence of biological origin.
- Rivers and streams if the water motion is sufficiently fast.
- Blood vessels, where for example they are one of the causes of heart tones observable with a stethoscope.

## Transition to turbulence

The transition to the turbulent regime can be, to some extent, predicted by the value of the Reynolds number, which quantifies the ratio of inertia forces to viscous forces in a fluid subject to relative motion within it, due to different values of the velocity field at different points in what is known as the boundary layer in the case of a surface bounding the fluid. Counteracting this effect is the viscosity of the fluid, which by increasing, progressively inhibits turbulence, as more kinetic energy is dissipated by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for certain types of flows, and is an indication of whether turbulent flow will be observed in a particular configuration.

This ability to predict the formation 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 rescaling fluid dynamics problems to exploit the dynamic similarity between two different cases of fluid flow, such as between a model aircraft and its full-size version. This similarity is not always linear, and applying Reynolds numbers to both situations allows the correct factors to be calculated. A flow in which kinetic energy is absorbed significantly due to the action of the fluid’s molecular viscosity gives rise to a laminar flow regime. Therefore, the dimensionless quantity of Reynolds number (Re) is used as a guide.

To recap:

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

## Differences between turbulent flow and laminar flow

A turbulent flow differs from a laminar flow because inside it there are vortical structures of different size and speed that make the flow not predictable in time even if the motion remains deterministic. That is, the motion is governed by the laws of deterministic chaos: if we were able to know “exactly” all the velocity field at a given time and we were able to solve the Navier-Stokes equations we could get all the future motion fields. But if we know the field with a very small inaccuracy, this after a certain time would make the solution found completely different from the real one.

For example, in the case of motion in a cylindrical duct, in case of turbulent regime the fluid moves in a disordered way, but with an average speed of advance almost constant on the section. In the case of laminar motion instead the trajectories are straight and the velocity profile is parabolic or Poiseuille. The Reynolds number for which the transition from laminar to turbulent regime occurs in this case is Re = 2300. However, this value is strictly dependent on the amplitude of disturbances present in the flow before the transition to the turbulent regime. Therefore, it is theoretically possible to obtain laminar flows for higher values of the Reynolds number.

Turbulent motions possess peculiar characteristics such as:

**Irregularity**. Turbulent flows are always highly irregular. For this reason, turbulence problems are usually treated statistically rather than deterministically. A turbulent flow is chaotic, but not all chaotic flows can be called turbulent.**Diffusivity**. The large availability of kinetic energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic responsible for increasing mixing and increasing the rate of transport of mass, momentum, and energy in a flow is called diffusivity. Turbulent diffusion is usually parameterized by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with molecular diffusivities, but has no real physical meaning, being dependent on the particular characteristics of the flow, and not a true property of the fluid itself. Furthermore, the concept of turbulent diffusivity assumes a constitutive relationship between turbulent flow and the gradient of an average variable similar to the relationship between flow and gradient that exists for molecular transport. At best, this assumption is only an approximation. However, turbulent diffusivity turns out to be the simplest approach for the quantitative analysis of turbulent flows and many models have been postulated to calculate it. For example, in large bodies of water such as oceans this coefficient can be found using Richardson’s four-thirds power law and is governed by random walk principles. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder’s formula.**Rotationality**. Turbulent flows possess non-zero vorticity and are characterized by an important mechanism of three-dimensional vortex generation known as vortex stretching. In fluid dynamics, these are essentially vortices subject to “stretching” associated with an increase in the component of vorticity in the stretching direction due to conservation of angular momentum. Furthermore, vortex stretching is the fundamental mechanism upon which the turbulent energy cascade relies to establish and maintain an identifiable structure function. In general, the stretching mechanism involves thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of the fluid elements. As a result, the radial length scale of the vortices decreases and the larger structures in the flow break down into smaller structures. The process continues until the small-scale structures are small enough to transform their kinetic energy into heat due to the molecular viscosity of the fluid. According to this definition, turbulent flow in the strict sense is therefore always rotational and three-dimensional. However, this definition is not shared by all scholars in the field, and it is not uncommon to come across the term two-dimensional turbulence, which is rotational and chaotic, but lacks the vortex stretching mechanism.**Dissipation**. In order to maintain a turbulent flow, a persistent source of energy is required because turbulence dissipates it rapidly, converting kinetic energy into internal energy by viscous shear stresses. In the turbulent regime, the formation of vortices having widely varying length scales occurs. Most of the kinetic energy of turbulent motion is contained in large-scale structures. Energy “precipitates” from these large-scale structures to smaller-scale structures by a mechanism due to purely inertial effects and essentially invisible. This process continues, creating smaller and smaller structures thus corresponding to a hierarchy of vortices. At the end of this process, structures are formed that are small enough to be affected by the effects due to molecular diffusion, and then viscous energy dissipation takes place. The scale at which this occurs is the Kolmogorov scale. This process is referred to as an energy cascade, for which the turbulent flow can be modeled as a superposition of a spectrum corresponding to the vortices and fluctuations of the velocity field i on an average flow. Vortices can be in a sense defined as coherent structures of the velocity, vorticity, and pressure fields. Turbulent flows can be viewed as consisting of an entire hierarchy of vortices over a wide range of length scales, and this hierarchy can be described quantitatively by the energy spectrum, which measures the kinetic energy of the velocity fluctuations for each length scale (each corresponding to a wave number). The scales in the energy cascade are generally unpredictable and highly non-symmetrical.

## History

Already Leonardo da Vinci drew and commented particular turbulent flows. The first scientific experiments carried out on this type of motions are due to an experiment conducted by Osborne Reynolds in 1883 at the hydraulics laboratories of the University of Manchester. Reynolds’ experiment consisted in the visual observation of the turbulent motions of a tube of a liquid thread distinguished from the surrounding liquid by a dye. For this purpose Reynolds had set up a transparent tube immersed in a tank full of water, which fed the tube. The mouth of the tube had been carefully shaped to prevent turbulence at the inlet.

The experiment was conducted with three different pipes. Reynolds observed that for relatively low fluid velocities the motion was quite regular and the thread was almost straight. Repeating the experiment at higher velocities, Reynolds observed confined perturbations migrating downstream. As he increased the velocity further, he observed that the color of the thread tended to spread throughout the pipe. Reynolds identified the condition in which the liquid fillet blended with the surrounding liquid as the “turbulent regime.”

Repeating the experiment with other pipes, Reynolds identified a critical value of the following dimensionless number, which would be named Reynolds number in his honor. For values of the Reynolds number greater than about 4000 the motion was presented as turbulent, while for values less than 2000-2500 the motion was presented as laminar.

The main difficulty in studying turbulence is the simultaneous presence of a large number of vortical structures of different characteristic magnitude, called “vortices”. Moreover, all these characteristic structures interact mutually with each other due to the nonlinear structure of the Navier-Stokes equations. All these peculiarities make the classical analytical approach difficult to apply.

In 1922 Lewis F. Richardson introduced the concept of energy cascade, while in 1941 the first statistical theory of turbulence was developed by the Soviet mathematician and physicist Andrej N. Kolmogorov.

In recent decades, the study of turbulence has made great strides forward, both for an advancement of technologies used in experimental studies, but especially by the introduction of computer simulations, which allow to study quantitatively in detail turbulent flows through the numerical integration of the Navier-Stokes equations. In particular, in the study of the most fundamental aspects of turbulence, great use is made of direct numerical simulations (DNS), which contrary to techniques more widely used in engineering as the Navier-Stokes equations averaged (RANS) or the Large Eddy simulations (LES), do not require additional assumptions on the behavior of the flow, since they explicitly solve all spatial and temporal scales of the system. However, the price to pay is a very high computational cost, so that even the numerical study of the simplest flows requires the use of powerful supercomputers.

## Extensions of the concept of turbulence

Over time, physicists have begun to use the term turbulence to refer to a number of phenomena related to that of fluid turbulence in the narrow sense. A first extension is that of magnetohydrodynamic turbulence, which occurs in electrically conducting fluids, such as plasmas. In this case, in addition to the typical effects of ordinary fluids (such as inertial forces, viscosity or gravity), electric and magnetic fields must also be taken into account, so their dynamics will not be described by the Navier-Stokes equations, but by those of magnetohydrodynamics (MHD). MHD turbulence phenomena assume great importance in astrophysics (e.g., in the dynamics of stellar atmospheres or accretion disks), or in the design of nuclear fusion reactors.

In condensed matter physics, quantum turbulence in superfluids is observed. This is a phenomenon that occurs at temperatures close to absolute zero in fluids such as helium-4, and differs from usual turbulence in several respects, such as the complete absence of viscosity, or the fact that the circulation associated with a vortex is no longer a continuous quantity, but a quantized one. The quantum vortices observed in superfluids are very similar to the magnetic field vortices that occur in superconductors. The fact that superfluids could give rise to turbulent motion was first predicted by Richard Feynman in 1955.

A different phenomenon is that of elastic turbulence, first observed in 2000. It was found that polymeric solutions could give rise to chaotic flows even in situations where one would expect laminar flow (e.g., for low Reynolds number values), and such a regime was given the name elastic turbulence, since such flows have similarities to true turbulent flows, such as a kinetic energy spectrum that follows a power law, or a noticeable increase in diffusivity. Since the equations describing such solutions can, to some extent, be rewritten in a form analogous to those of magnetohydrodynamics, it is possible to observe phenomena analogous to those in plasmas, such as “elastic Alfven waves”.

A more generalized extension of the concept of turbulence is that of wave turbulence, in which a physical system far from equilibrium gives rise to a set of nonlinear waves interacting with each other in a chaotic manner, resulting in phenomena analogous to those in fluid turbulence, such as the energy cascade. This phenomenon can be observed in fluid dynamics (for example in sea waves), but also in plasmas, in nonlinear optics or in gravitational waves, that is in general in all media with nonlinear dispersion relation.

## References

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- R.P. Feynman, Application of quantum mechanics to liquid helium, in II. Progress in Low Temperature Physics, vol. 1, Amsterdam, North-Holland Publishing Company, 1955.
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