The significance of coherent flow structures for the turbulent mixing ...

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5 Investigation of the xy-plane
5.1 The statistical description of turbulence
The most striking characteristics of all turbulent flows is the complex spatio-temporal variation
of the fluid mechanical variables. Even under stationary boundary conditions they fluctuate
without any apparent regularity with different amplitude, wavelength and orientation [76].
From the present point of view the details of turbulent flows seem to be neither predictable
nor reproducible because the finest details of the flow depend very much on the initial and
boundary conditions, which are never known exactly, and in addition they are strongly affected
by instabilities due to small disturbances. However, as the finest details of the turbulent motion
are only of academic interest, because no serious practical application is conceivable where all
details of the flow are required, the majority of information being present in a turbulent flow
field can be reduced to a treatable form by applying the formalism of statistical mechanics [23,
64, 65]. The strength of this approach is well established by the kinetic theory of gases, where
the effect of the molecular motion on the macroscopic fluid can be described by the kinematic
viscosity é , provided the intermolecular distance ê m, given by the mean free path length of
the molecular gas, is much smaller than the Kolmogorov micro-scales ê k which represent the
wavelength of the smallest macroscopic eddies, whose size depends only on the total energy ë
and the viscosity.
(5.1)
ê m ì
ê k صí
In this approximation, the macroscopic fluid motion, represented by a fluid element, is considered
as a large scale perturbation of the molecular chaos which becomes visible when the
molecular motion is approximated by a continuous velocity field. In principle, this can be done
by averaging the velocities of all molecules in the neighbourhood of a particular position. The
basic feature of the statistical approach to turbulence is the transition from the consideration of
a single turbulent flow to the consideration of the statistical ensemble of all similar flows. Each
flow supplies a unique realization of the random variables õ (velocity , ö pressure , ÷ density
and ø temperature ) and each particular result represents one ù of possible results which might
have occurred. For the characterisation of the physical phenomenon, the statistical theory supplies
basically four quantities, namely the mean value, the variance, the probability density
function and the auto-correlation function of the random variable. The mean value, defined in
equation (5.2), can be seen as a static measure of the intensity of the signal and the variance,
defined in equation (5.3), possesses information concerning the dynamics of the random signal
under investigation. The probability density function, defined in equation (5.4), furnishes
information about the properties of the data in the amplitude domain and the auto-correlation,
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