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Chapter 1
A Brief Introduction to Turbulence
The scope of this chapter is to recall some of the bases of the turbulence theory and its statistical analysis. The emphasis is put on turbulent flow features that are of primary interest for turbulent flow prediction and modelling: turbulent scales, Kolmogorov cascade, coherent structures in shear flows, turbulence production and dissipation. The reader interested in more in-depth discussions of the subject is referred to several reference textbooks: [Chassaing, 2000; Davidson, 2004; Hinze, 1959; Lesieur, 1990; Piquet, 1999; Tennekes and Lumley, 1974; Sagaut and Cambon, 2008].
1.1 Common Features of Turbulent Flows
1.1.1 Introductory concepts
Most fluid flows occurring in nature as well as in engineering applications are turbulent. Consequently, it does not take any further comment to emphasize that numerical simulations of turbulent flows are of outstanding importance for the scientific as well as for the engineering community. Even though many turbulent flows can be easily observed, it is very difficult to give an accurate and accepted definition of turbulence. However, researchers and engineers generally agree on some characteristics of turbulent flows. For this purpose, let us observe what happens in the turbulent flow past a sphere (see Fig. 1.1) and list the most generally agreed features.
ā¢ Unpredictability
The irregularity of the flow downstream separation makes a deterministic description of the motion detailed as a function of time and space coordinates impossible. Randomness is clearly shown in the above figure and is a characteristic of all turbulent flows. This explains why statistical methods are often considered.
Fig. 1.1 Flow past a sphere at
. Note its continuum of scales from large to small ones which is one of the most fundamental aspects of turbulent fluid flows. Photograph by H. WerlƩ. Courtesy of J. DƩlery, ONERA, France.
ā¢ Three-dimensionality of the vorticity fluctuations
The flow past the sphere is obviously three-dimensional and highly unsteady. Note that the shear layer emanating from the separation line on the cylinder is a region of strong coherent vorticity. In general, vorticity dynamics plays an important role in the analysis of turbulent flows.
ā¢ Diffusivity
Spreading of velocity fluctuations becomes stronger as the distance from separation increases. The diffusivity of turbulence is one of the most important properties as far as engineering applications are concerned (mixing enhancement, heat and mass transfer).
ā¢Broad spectrum
Turbulent fluctuations occur over a wide range of excited length and time scales in physical space leading to broadband spectra in wave number space.
In his book, Hinze [Hinze, 1959] suggests that:
to describe a turbulent motion quantitatively, it is necessary to introduce the notion of scale of turbulence: a certain scale in time and a certain scale in space.
In other words, turbulence is a multiscale problem with a highly non-linear coupling between these scales. This picture illustrates why the accurate prediction of turbulent flows is such a difficult problem.
1.1.2 Randomness and coherent structure in turbulent flows
Though turbulent flows exhibit broadband spectra, there is evidence from Fig. 1.1 that high Reynolds number turbulent flows are far from being totally disorganized. What strikes us when looking at the shear layer downstream separation is the roll-up of eddies which can be found downstream with approximatively the same shape. Such eddies preserving a certain spatial organization are called coherent structures and retain their identity for much longer times than the eddy turn-over time characteristic of the turbulent fluctuation.
There are still controversies regarding the definition of coherent structures (see [Haller, 2005]) but there is general agreement on their existence and importance in the transport and mixing phenomena. Therefore the identification of coherent vortices plays an important role in the analysis of turbulent flows. Most common definitions are associated with vortical motion (see [Sagaut and Cambon, 2008] for an exhaustive review) and turbulent structures are often exhibited by showing a positive iso-value of the criterion Q [Hunt et al., 1988]. Vortex tubes are defined as the regions where the second invariant of the velocity gradient tensor Q is positive:
where
is the vorticity tensor and
is the rate of strain tensor.
Examples of coherent structures include for instance the horseshoe vortices observed in turbulent boundary layers (see Fig. 1.2) and mixing layers (see Fig. 1.3), and the vorticity tubes (often called filaments or worms) observed in statistically homogeneous flows (see Fig. 1.4).
Fig. 1.2 Wind-tunnel visualization of large-scale structures in the outer layer of a turbulent boundary layer. Courtesy of M. Stanislas, IMFL, France.
Turbulent flows are neither determi...
