CHAPTER 6 : SPECTRAL DYNAMICS
Spectra are decompositions of a function into waves of different wavenumbers and, thus, give
us an opportunity to think about the way in which waves, or eddies, of different sizes
exchange energy with each other. We know from previous chapters that turbulence
commonly receives its energy at large scales, while the dissipation of energy occurs at very
small scales. We also know that there often exists a range of eddy sizes which are not directly
affected by the energy maintenance and dissipation mechanisms, which is called inertial
subrange.
Experimental measurements may be made along a straight line at a fixed time, or at a fixed
point as a function of time, generating a random function of position or time, respectively. The
transform variable (of Fourier transformation) is a wavenumber or frequency accordingly.
Spectra obtained in this way are called one-dimensional spectra because the measurements
producing them were taken in one dimension.
Since turbulence is three dimensional, one-dimensional spectra may not be very useful for the
description of turbulence. Indeed, one-dimensional spectra may be misleading in three
dimensional fields. Assume that there exist a wave field of wavenumber , and that we are
making measurement in the dircetion shown in the figure below which is oblique to the
direction of propogation.
Direction of
Propogation
Wavefronts
2
'
2

Direction of measurement
What we perceive will then be a wavenumber of ’ which is larger than the actual value of .
This is called aliasing. This can be avoided if we could make measurements in all possible
directions.
Often the velocity components, say, u1, u2, u3, are measured seperately. Since the spectrum
represents all of the kinetic energy (what we refer by spectra is indeed power spectra) at a
given wavenumber, we add the spectra of u1, u2, u3 together, which is referred as threedimensional spectrum. Let’s show that integral of power spectrum indeed represents the
kinetic energy per unit mass.
The correlation tensor


 
R ij r   u i x, t  u j x  r , t 
can be used to define spectrum tensor (see also previous chapter)


 ij k 

 
 
exp  i   r  R  r  d r



2
1
ij
3

or as Fourier-pair


 
 
R ij r      exp  i   r   ij   d

so that
R ii o   u i u i   

 
   d
ii

On the other hand, E () (3-D spectrum) was defined as sum of one-dimensional spectra over
all possible directions
E   
1
 ii   d
2 
where d is an infenitesimal area on a spherical shell of radius , and ½ factor is put to make
integral of 3-D spectrum equal to the kinetic energy per unit mass.


1
0 Ed  2 0




 
1
 ii d d  2  ii d  R ii 0  u i u i
 E d  2 u u
1
i
i
= kinetic energy per unit mass
0
Spectral Definiton of an Eddy
The Fourier transform of a velocity field is a decomposition into waves of different
wavelengths; each wave is associated with a single Fourier coefficient. An eddy, however, is
associated with many Fourier coefficients and phase relations among them. Fourier
transforms are used to define an eddy, because they are convenient to measure. If we want
to decompose a velocity field into eddies instead of waves, we need more sophisticated
transforms such as wavelet transforms.
E ()
0.1 
0.6 

Figure 2
1.6 
ln 
It is convenient to define an eddy of wavenumber, , as a disturbence containing energy
between a certain interval, as for example 0.62  - 1.62  as shown in Figure 2.
This choice centers energy around  on a logarithmic scale and makes the width of the
contribution to the spectrum equal to . Such an eddy will have a variation in parameter
domain (space or time) as shown in Figure 3.
FIGURE 3
This is the Fourier transform of previous figure where we had a narrow band around . Now
we have in the above figure a wave of wavelength
2
with envelope whose width is inverse

of the bandwidth. Since the bandwidth in the previous figure was , the width of the envelope
of the eddy is of order of 1/.
Local Isotropy and Equilibrium of Small Eddies :
An important aspect of the turbulent flow depend on the energy-containing eddies in that the
rate of energy dissipation in turbulent flow is determined by the structure and intensity of
these eddies. Since the actual conversion of mechanical energy to heat is carried out by
much smaller eddies of almost negligible total energy, there is a strong suggestion that the
energy is transferred to them by a cascade of instabilities, larger eddies breaking up to form
eddies one size smaller that become unstable in turn and break up into still smaller ones. If
the rate of energy flow down the cascade of eddy size is limited by the capacity of the first
instability, the smaller eddies must adjust their motion to pass on the imposed energy flow to
the smallest eddies that dissipate it as heat. If the Reynolds number of the flow is sufficiently
large, the smaller eddies must be in the state of absolute equilibrium in which the rate of
receiving energy from larger eddies is very nearly equal to the sum of the rates of loss to
smaller eddies by breaking up and by working against viscous stresses. Further the motion of
the smaller eddies is related only weakly to the inhomogeneous and anisotropic large eddies
and their motion should be nearly isotropic. The theory of local isotropy asserts that the
motion of the smaller eddies depends only on the energy flow from the energy-containing
eddies and on the fluid properties, otherwise being the same for all kinds of turbulent flows.
Therefore, there is no need to study the smaller eddies in all kinds of turbulent flow. Once the
energy loss from the large energy-containing eddies is known, the structure of the smaller
ones is determined.
The theory of local isotropy applies to the small scale aspects of the turbulent motion, with
eddies that are too small to contribute more than a small fraction to the total kinetic energy but
are responsible for nearly the whole of the energy dissipation by viscous action. If the
spectrum function E() is used to define the energy associated with eddies of size -1, the
theory applies to wave numbers greater than e where e is the smallest wavenumber
satisfying the inequalities :


 Ed   Ed  q
1.
e
0
e
  Ed
2
2.
2
0



  Ed  V
2
0
The first condition is satisfied if e is large compared with the centroid of the spectrum
function, in practice with -1 where  is the dissipation length scale. The second condition
requires that e should be small compared with the centroid of the dissipation spectrum
function, 2 E(). Both condition can be satisfied if Reynolds number of the turbulent motion
q 
2
1/ 2

V
is large.
The spectral range,  > e is nearly in a state of absolute equilibrium in the sense that all parts
of it can adjust themselves to external changes at rates that are large compared with rates of
change of the whole flow. Then the structure of the equilibrium range of eddies is dependant
only on the driving forces and fluid properties. The driving forces have their origin in the mean
flow and in the eddies outside the equilibrium range, but if the energy reaching the range has
passed through many intermediate stages of instability and breakdown, it is likely that the
motion will be nearly independent of the directional bias and anisotropy of the large-scale
motion. Then the spectrum function is completely determined by the scalar, E() which can
depend only on the energy flow into the range and on the fluid properties. In such an inertial
subrange of the equilibrium spectrum,
E  C  2 / 3  5 / 3
where C is an absolute constant of order 1. In brief, the above inertial form of the spectrum
holds for eddies small compared with the energy-containing eddies, but large compared with
those responsible for viscous dissipation of energy.
E ()
3
-5
Viscous dissipation
subrange
Inertial Subrange

The above law is called Kolmogorov’s –5/3 decay law and is universal within inertial
subrange.
No strong predictions can be made about the form of the spectrum function for wave-numbers
in the viscous subrange.