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evolving in time. This assumption is valid when the mean flow is large compared with
the r.m.s. (root mean square) velocity of the fluctuations. The spatial and time scales
(l and
D
t) are then simply related through advection in the mean flow at speed U, i.e.
2p
2pU
l
¼
t
¼
l
=
U
;
o
¼
t
¼
kU
:
ð
4
:
64
Þ
Using this relation, we can convert
(k).
Many different types of turbulence velocity spectra can be defined in terms of
different combinations of velocity components depending on wavenumber or fre-
quency variables. For our purpose it will be sufficient to use the scalar energy
spectrum
C
(o) to the wavenumber spectrum
C
E
S
(k) which expresses the distribution of TKE at different scalar wave
numbers k. In other words, within a wavenumber interval dk, centred on wave
number k, there is an amount of energy per unit mass
E
S
(k)dk. Summing over all
k gives the total TKE:
ð
1
0
E
S
ð
1
2
ð
u
0
2
v
0
2
w
0
2
k
Þ
dk
¼
þ
þ
Þ:
ð
4
:
65
Þ
E
S
(k), if we can determine it, tells us how energy is distributed over different spatial
scales regardless of direction.
The Equilibrium Spectrum
In an important advance in turbulence theory, Kolmogorov argued that, in a steady
state,
E
S
(k) should have a general equilibrium form for part of the wavenumber
range. He did this on the basis of a bold hypothesis about how energy is transferred
through the spectrum, from the largest eddies (wave number
k
0
) which contain
most of the energy, to small scales. He postulated that the energy transfer occurs by a
large number of short steps from one spatial scale to an adjacent smaller scale, as in
the schematic picture of
Fig. 4.15
.
∼
k
0
k
ν
E
(
k
)
ε
0
k
Large eddies
Smaller eddies
Smaller eddies
Dissipation
“ Order ”
Increasing disorder
Homogeneous & isotropic
Figure 4.15
Schematic of energy transfer in the spectrum of turbulence. In the Kolmogorov
cascade, the energy is transferred from the large scales of the mean flow via eddies of
decreasing size to the smallest scales where it is dissipated as heat.
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