Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

14 1. Newtonian turbulence
of the correlation function of order n involves that of order n + 1; it will be then
necessary to make assumptions on the statistics of the velocity field in order to
close the equations of motion for correlation functions. Despite the importance of
the problem, instead of concentrating on the study of closure approximations, in
the following we will focus on scaling properties which can be inferred by means
of phenomenological arguments.
Let us end this section introducing an important length scale that can be formed
from the correlation function. If uL is the longitudinal (parallel to the separation
vector r) velocity component, then the longitudinal correlation coefficient is defined
as:
f(r) ≡ 〈uL(x)uL(x + r)〉
u2 (1.40)
rms
where < u2 i >= u2rms ∀i, due to isotropy. Suppose that we expand f(r) about
r = 0. Then, recalling that f must be a symmetric function of r, we can write
where we defined the Taylor scale λ this way:
f(r) = 1 − r2
2λ 2 + O(r4 ) (1.41)
1
λ 2 ≡ −f ′′ (0) (1.42)
that is, by fitting a parabola to the longitudinal correlation function for small values
of r.
It is possible to show that the Taylor scale can be equivalently defined as
1
λ2 ≡ 〈(∂xux) 2 〉
〈u2 x〉
(1.43)
Under isotropy we have: u 2 rms = 2E/3 and ɛ = 15ν < (∂xux) 2 >, so that the
definition (1.43) of λ implies the relation
λ 2 = 15ν u2 rms
ɛ
= 5E
Z
(1.44)
between the Taylor scale and global quantities such as the energy dissipation rate
ɛ, the kinetic energy E and the enstrophy Z.
With the new length λ, the Taylor-scale Reynolds number can be constructed:
Rλ ≡ urmsλ
ν = √ 15Re (1.45)
which is commonly used for experimental data, because it is easier to measure
than the integral scale Reynolds number defined in section 1.1.1.
14