As shown by the global energy balance (Eq. 1.16) the
non-linear term in Navier-Stokes equation does not change
the total kinetic energy. Nevertheless it plays a fundamental
role in turbulence, because it is responsible for the energy
transfer between different modes which is the origin of the
turbulent cascade. To describe how it is involved in the
energy transfer it is worthwhile to consider the energy
balance in Fourier space. For the sake of simplicity
we will consider the infinite volume limit, in which the fluid
is supposed to fill the entire space, and the Fourier transform reads
Using the incompressibility and the symmetry of the integrals for
it is possible to rewrite Eq. (1.20) as
Let's now introduce some notations.
The two-point correlation function is defined as
The energy spectrum is defined as the integral
of the square modulus of velocity over a shell
with fixed modulus in Fourier space:
The temporal derivative of the two-point correlation function
is obtained from Navier-Stokes equation as
The energy balance is obtained from Eq. (1.33) remembering
the relation (1.32) between the energy spectrum and the
two-point correlation function. Using the antisymmetry
and the reality condition
on gets
Defining the enstrophy spectrum as:
If the external forcing is a Gaussian process
-correlated in time, whose statistic is determined by the
correlation
, the input of
energy is flow-independent, i.e, the injection energy spectrum
is uniquely determined by the statistics of the forcing.
In the case of a large scale forcing,
with a forcing correlation length
such that
Equation (1.24) for velocity involves the
two-point correlation function, and Equation (1.33)
for the two-point correlation function requires the tree-point one.
It is easy to understand that the presence of a quadratic
term in Navier-Stokes equation reproduces this closure problem
at every order, i.e. the equation for the -point
correlation function will
require the
one. During the last fifty years several
closures have been proposed, i.e. assumptions on the statistics
of velocity which allow to obtain a closed set of equations
for the correlation functions, from the simplest Quasi-Normal closure
in which the fourth-order moments of velocity distribution
are expressed in term of the second-order ones, in the same way
of what happens for a Gaussian variable, to the
Eddy-Damped-Quasi-Normal closure proposed by Orszag [8].