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Steepening of the energy spectrum
In absence of friction the requirement of a
constant enstrophy flux in the direct cascade allows to obtain
a dimensional prediction for the energy spectrum
(see Chapter 1).
As already shown by Bernard [33] and Nam et al [30],
a non-vanishing friction regularizes the flow depleting
the formation of small-size structures
and results in a steeper spectrum [30]
 |
(2.5) |
In the range
the exponent
coincides with the scaling exponent
of the second-order moment
of vorticity fluctuations
. An explicit expression for the
correction
to the spectral slope will be given in
Section. 2.4.
The physical origin of the steepening of the spectrum is clear:
part of the enstrophy which is transported to small scales is removed
by the friction during the cascade process,
thus the amplitude of fluctuations which reach the small scales is reduced.
Since the energy spectrum is steeper than
,
the second-order velocity structure function
is dominated by the IR contribution of the spectrum and
trivially displays smooth scaling independently of the value of
.
Thus the presence of a non-vanishing friction term ensures
that the velocity fields in the direct enstrophy cascade is smooth.
Next: Analogies with the passive
Up: Effects of friction in
Previous: Origin of the friction
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Stefano Musacchio
2004-01-09