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# Turbulence

In fluid dynamics, turbulence or turbulent flow is a 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. Flow that is not turbulent is called laminar flow. The (dimensionless) Reynolds number characterizes whether flow conditions lead to laminar or turbulent flow; e.g. for pipe flow, a Reynolds number above about 4000 (A Reynolds number between 2100 and 4000 is known as transitional flow) will be turbulent. 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 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 changes, sometimes resulting in a reduction of overall drag. Because 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.

Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large scale structures. The energy "cascades" from these large scale structures to smaller scale structures by an inertial and essentially inviscid 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.

In two dimensional turbulence (as can be approximated in the atmosphere or ocean), energy actually flows to larger scales. This is referred to as the inverse energy cascade and is characterized by a k − (5 / 3) in the power spectrum. This is the main reason why large scale weather features such as hurricanes occur.

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.

When designing piping systems, turbulent flow requires a higher input of energy from a pump (or fan) than laminar flow. However, for applications such as heat exchangers and reaction vessels, turbulent flow is essential for good heat transfer and mixing.

While it is possible to find some particular solutions of the Navier-Stokes equations 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. Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the notion of the cascade (energy flow through scales) and self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified [1]. Still, the complete description of turbulence remains 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."[2] A similar witticism has been attributed to Horace Lamb (who had published a noted text book on Hydrodynamics)—his choice being quantum mechanics (instead of relativity) and turbulence. Lamb was quoted as saying 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 optimistic."[3]

## Contents

### Examples of turbulence

• Smoke rising from a cigarette. For the first few centimeters, the flow remains laminar, and then becomes unstable and turbulent as the rising hot air accelerates upwards. Similarly, the dispersion of pollutants in the atmosphere is governed by turbulent processes.
• 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 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.
• The mixing of warm and cold air in the atmosphere by wind, which causes 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, etc.) and machines (for instance, internal combustion engines and gas turbines).
• The external flow over all kind 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.
Unsolved problems in physics: Is it possible to make a theoretical model to describe the behavior of a turbulent flow — in particular, its internal structures?
• Race cars unable to follow each other through fast corners due to turbulence created by the leading car causing understeer.
• Trucks. In windy conditions and/or on the motorway your vehicle gets buffeted by their wake.
• Round bridge supports under water. In the summer when the river is flowing slowly the water goes smoothly around the support legs. In the winter the flow is faster, so a higher Reynolds Number, so the flow may start off laminar but is quickly separated from the leg and becomes turbulent.

## References

1. ^ http://www.weizmann.ac.il/home/fnfal/KRSPhysTodayApr2006.pdf
2. ^ http://www.eng.auburn.edu/users/thurobs/Turb.html Turbulence
3. ^ http://www.fortunecity.com/emachines/e11/86/fluid.html Turbulent Times for Fluids. It's important to notice that turbulence is completely a different case from instability.
• Falkovich, Gregory and Sreenivasan, Katepalli R. Lessons from hydrodynamic turbulence, Physics Today, vol. 59, no. 4, pages 43-49 (April 2006).[1]
• U. Frisch. Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, 1995.[2]
• T. Bohr, M.H. Jensen, G. Paladin and A.Vulpiani. Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.[3]

### Original scientific research papers

• Kolmogorov, Andrey Nikolaevich (1941). "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers". Proceedings of the USSR Academy of Sciences 30: 299-303. (Russian), translated into English by Kolmogorov, Andrey Nikolaevich (July 8 1991). "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers". Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 434 (1890).
• Kolmogorov, Andrey Nikolaevich (1941). "Dissipation of energy in locally isotropic turbulence". Proceedings of the USSR Academy of Sciences 32: 16-18. (Russian), translated into English by Kolmogorov, Andrey Nikolaevich (July 8 1991). "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers". Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 434 (1890): 15-17.