It is assumed that any length scale derived in this paper (there are three) depends only on the dynamical conditions of the fluid and not on a constitutive property of the fluid (such as thermodynamic conductivity). Prandtl's mixing length comes from a Lagrangian thought process and is the turbulent analog of the mean free path of molecules in the Kinetic Theory of Gases. In fact, up until this paper, most turbulence theory had only been discussed in a Lagrangian mind set. An interesting result of the theory in this paper is the first treatment of turbulence theory in both a Lagrangian and Eulerian frame of reference. This had to have made the life of experimentalists much easier since measurements are much easier accomplished in an Eulerian space: comparing two points in space at the same instant in time is straightforward.
Without going into the paper entitled `Diffusion by Continuous
Movements', Taylor invoked the main result where he proved that
the mean square distance diffused by a particle (in a Lagrangian
sense), 
, is related to the time correlation of the particle
If the time considered is large such that there is a time T
where the time correlation goes to zero, then for t>T, it can
be said that from equation 
Under these circumstances, there exists a (Lagrangian) length scale such that
To consider the problem in an Eulerian manner, Taylor defined
another type of correlation function. If 
is the spatial
correlation (of the velocity at two points a distance y apart)
and is large in an eddy and approaches zero outside the eddy (say
a distance Y), then a plot of verses y will give the
statistical distribution of the velocity in a flow field. It is
this type of correlation function which is so widely used in the
field of turbulent statistics today.
An Eulerian length scale can be determined using the spatial
correlation in a manner similar to the Lagrangian length
scale.
This length scale is also commonly referred to as the integral length scale of the flow. It is a measure of the average spatial extent or coherence of the fluctuations. For a particular point in the flow, the magnitude of the integral length scale is a function of the not just the quantity correlated but on the direction of separation as well depending on which directional correlation is used. It is also the length scale of the largest eddy or the width of the flow.
A third length scale, 
, is developed using this Eulerian
thought process and is called the Taylor microscale. Taylor
incorrectly postulated that it is the ``average size of the
smallest eddies" which are responsible for the dissipation of
energy by viscosity. Since its derivation appears in the
discussion of energy dissipation, it will not be discussed in
this context.
As an aside, the true size of the smallest eddy was later deduced
by Kolmogorov and is thus named the Kolmogorov scale,
. By definition, Kolmorogorov's
length and velocity scales ( 
) result in a
Reynolds number equal to 1. Thus, the assumption is that the
smallest-scale motion of turbulence is quite viscous and that the
viscous dissipation adjusts itself to the energy supply by
adjusting length scales.
