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THE MODELING OF TURBULENCE
Most fluid motion that occur in nature is turbulent. Typical examples include the boundary layer growing over an air-plane, the flow of natural gas in pipelines or the motion of interstellar gas clouds. This review is meant to introduce the reader to this exciting phenomenon. First, the nature of turbulence will be described, with a particular emphasis on the physics of the problem. Next, the equation governing the behavior of a turbulent flow will be derived. The concept of Reynolds averaging will also be introduced, leading to the well-known closure problem. Various Reynolds-Averaged Navier Stokes (RANS) models will be analyzed, ranging from the basic algebraic models to the two-equation models. Eventually, the final section will review more advanced approaches such as Large Eddy Simulation (LES) and Direct Eddy Simulation (DNS). This report is merely an introduction. For a detailed physics-based description of turbulence, the reader is referred to H. Tennekes and J.L. Lumley (1972) [1]. The textbook written by D.C. Wilcox (2006) on turbulence modeling for CFD [2] is also recommended.


INTRODUCTION TO TURBULENCE
Identifying a precise definition of turbulence is a challenge in itself. Nevertheless, some basic observations can be made.
Disorder:
Turbulent flows exhibit a distinct lack of orderly behaviour. Deterministic methods are typically abandoned in favor of statistical approaches (averages, probabilities). It is important to keep in mind that despite their apparent randomness, turbulent flows follow certain rules and cannot be considered as “pure chaos".
Increased diffusivity:
As a flow becomes turbulent, the rate of mixing is greatly enhanced. This point is extremely important, since it affects the drag, heat transfer, and other fundamental characteristics of the problem. In a laminar flow, the properties are slowly diffused through the domain. In a turbulent flow, these properties are transported by the eddies, increasing significantly the mixing rate.
Vortex stretching:
There is no such thing as two-dimensional or steady turbulence. The vortex lines in the flow are non-parallel and can only been captured with a full three-dimensional unsteady approach.
Turbulence cascade:
Most of the kinetic energy of a turbulent flow is contained within its large structures. The energy is transferred progressively to the smaller scales, where it is eventually dissipated into heat through the action of molecular viscosity. By nature, turbulent flows are therefore dissipative. This turbulence cascade is illustrated in the famous poem by Lewis F. Richardson (1920).
Big whorls have little whorls
That feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity.
Flow history effects:
The large structures are conveyed within the flow, carrying the smaller eddies with them. For this reason, it is not possible to compute the properties of a flow at a given position, without considering its upstream history.

References

[1] H. Tennekes and J.L. Lumley. First course in turbulence. MIT press, 1972.

[2] D.C. Wilcox. Turbulence modeling for CFD, volume 3. DCW industries La Cañada, 2006.