Part 2 (Fully-developed turbulence)

The word “developed” has already been employed for the small-scale three-dimensional turbulence which appears in the mixing-layer experiments. Fully-developed turbulence is a turbulence which is free to develop without imposed
constraints. The possible constraints are boundaries, external forces, or viscosity. One can easily observe that the structures of a flow of scale comparable
with the dimensions of the domain where the fluid evolves cannot deserve to
be categorized as “developed”.
The same remark holds for the structures directly created by the external
forcing, if any. So no real turbulent flow, even at a high Reynolds number, can
be “fully developed” in the large energetic scales. At smaller scales, however, rbulence will be fully developed if the viscosity does not play a direct role in
the dynamics of these scales.
This will be true if the Reynolds number is high
enough so that an “inertial-range” can develop.
In the preceding experimental
examples of the jet and the mixing layer, one actually obtains fully-developed
turbulence at scales smaller than the large energetic scales and larger than the
dissipative scales. On the contrary, in the majority of grid-turbulence experiments, the Reynolds number is not high enough to enable an inertial range
to develop. The small three-dimensional turbulent scales of the Earth’s atmosphere and oceans, or Jupiter and Saturn, are certainly fully developed. But
the planetary scales of these flows are not, because of constraints due to the
rotation, thermal stratification and finite size of planets. In this monograph,
the term “developed” will mainly be used for three-dimensional flows, though
it could be generalized to some high Reynolds number two-dimensional flows
constrained to two-dimensionality by some external mechanism which does
not affect the dynamics of the two-dimensional eddies once created.
An interesting issue about the structure of fully-developed turbulence concerns the possibility of fractal or multi-fractal distributions. This problem has
been studied by Mandelbrot  and Frisch .
Finally, we stress that it is possible, for theoretical purposes, to assume
that turbulence is fully developed in the large scales also, when studying
a freely-evolving statistically homogeneous turbulence (without any mean
shear): there is in this case no external force or boundary action.

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part(1) : (Introduction to Turbulence in Fluid Mechanics)

Everyday life gives us an intuitive knowledge of turbulence in fluids: the smoke
of a cigarette or over a fire exhibits a disordered behaviour characteristic of
the motion of the air which transports it. The wind is subject to abrupt
changes in direction and velocity, which may have dramatic consequences for
the seafarer or the hang-glider.

During air travel, one often hears the word
turbulence generally associated with the fastening of seat-belts. Turbulence is
also mentioned to describe the flow of a stream, and in a river it has important
consequences concerning the sediment transport and the motion of the bed.
The rapid flow of any fluid passing an obstacle or an airfoil creates turbulence
in the boundary layers and develops a turbulent wake which will generally
increase the drag exerted by the flow on the obstacle (and measured by the
famous Cx coefficient): so turbulence has to be avoided in order to obtain better aerodynamic performance for cars or planes. The majority of atmospheric
or oceanic currents cannot be predicted accurately and fall into the category
of turbulent flows, even in the large planetary scales. Small-scale turbulence
in the atmosphere can be an obstacle towards the accuracy of astronomic observations, and observatory locations have to be chosen in consequence. The
atmospheres of planets such as Jupiter and Saturn, the solar atmosphere or
the Earth’s outer core are turbulent. Galaxies look strikingly like the eddies
which are observed in turbulent flows such as the mixing layer between two
flows of different velocity, and are, in a manner of speaking, the eddies of a
turbulent universe. Turbulence is also produced in the Earth’s outer magneto-sphere, due to the development of instabilities caused by the interaction of
the solar wind with the magnetosphere. Numerous other examples of turbulent flows arise in aeronautics, hydraulics, nuclear and chemical engineering,
oceanography, meteorology, astrophysics and internal geophysics.

It can be said that a turbulent flow is a flow which is disordered in time
and space. But this, of course, is not a precise mathematical definition. The
flows one calls “turbulent” may possess fairly different dynamics, may be
three-dimensional or sometimes quasi two-dimensional, may exhibit well organized structures or otherwise. A common property which is required of
them is that they should be able to mix transported quantities much more
rapidly than if only molecular diffusion processes were involved. It is this latter property which is certainly the more important for people interested in
turbulence because of its practical app locations : the engineer, for instance, is
mainly concerned with the knowledge of turbulent heat diffusion coefficients,
or the turbulent drag (depending on turbulent momentum diffusion in the
flow). The following definition of turbulence can thus be tentatively proposed
and may contribute to avoiding the somewhat semantic discussions on this
matter:
• Firstly, a turbulent flow must be unpredictable , in the sense that a small
uncertainty as to its knowledge at a given initial time will amplify so as to
render impossible a precise deterministic prediction of its evolution (a).
• Secondly, it has to satisfy the increased mixing property defined above (b).
• Thirdly, it must involve a wide range of spatial wave lengths (c).

Such a definition allows in particular an application of the term “turbulent”
to some two-dimensional flows. It also implies that certain non-dimensional
parameters characteristic of the flow should be much greater than one: indeed,
let l be a characteristic length associated to the large energetic eddies of
turbulence, and v a characteristic fluctuating velocity; a very rough analogy
between the mixing processes due to turbulence and the incoherent random
walk allows one to define a turbulent diffusion coefficient proportional to lv.
As will be seen later on,l is also called the integral scale. Thus, if ν and κ are
respectively the molecular diffusion coefficients of momentum (called below the kinematic molecular viscosity) and heat (the molecular conductivity), the increased mixing property for these two transported quantities implies that
the two dimensionless parameters Rl = lv/ν and lv/κ should be much greater
than one. The first of these parameters is called the Reynolds number, and
the second one the Peclet number. Notice finally that the existence of a large
Reynolds number implies, from the phenomenology developed in Chapter 6,
that the ratio of the largest to the smallest scale is of the order of R3/4.